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Numerical holographic condensed matter Smyth, Darren 2016

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Numerical Holographic Condensed MatterbyDarren SmythB.Sc., National University of Ireland, Galway, 2009M.Sc., University of Waterloo, 2010A 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)February 2016© Darren Smyth, 2016AbstractThis thesis studies strongly coupled phases of condensed matter physics using acombination of gauge-gravity correspondence and numerical methods. We exam-ine holographic models of the condensed matter phenomena of: vortex formation inthe spontaneously broken phase of gauge theories, spontaneous breaking of trans-lational invariance by periodic modulation, properties of (non-)Fermi liquids, andmetal-insulator transitions in systems with sourced periodic modulation.In Chapter 2, we formulate a criterion for the existence of a Higgs phase basedon the existence of bulk solitons. This criteria is applicable when the microscopicdetails of the gauge theory are unknown. We demonstrate the existence of suchsolitons in both top-down and bottom-up examples of holographic theories andexamine their thermodynamics.In Chapter 3, we construct inhomogeneous, asymptotically Anti-deSitter Space(ADS) black hole solutions in Einstein-Maxwell-axion theory corresponding to thespontaneous breaking of translational invariance and the formation of striped orderin the dual 2+ 1 dimensional Quantum Field Theory (QFT). We investigate thephase structure as function of parameters.In Chapter 4, we continue the study begun in Chapter 3. On domains of bothfixed and variable wavenumber, we find a second order phase transition to thestriped solution in each of the grand canonical, canonical and microcanonical en-sembles. We also examine the properties of the bulk black hole solutions.In Chapter 5, we consider a phenomenological model whose bosonic sectoris governed by the DBI action, and whose charged sector is purely fermionic. Inthis model, we demonstrate the existence of a compact worldvolume embedding,stabilized by a Fermi surface on a D-brane. We study the bulk and dual QFT ther-iimodynamic and transport properties.In Chapter 6, we analyze low energy thermo-electric transport in a class ofbottom-up, holographic models in which translation invariance has been broken.As a function of our choice of couplings, which parameterize this class of theo-ries, we obtain (i) coherent metallic, or (ii) insulating, or (iii) incoherent metallicphases. We use a combination of analytical and numerical techniques to studythe Alternating Current (AC) and Direct Current (DC) transport properties of thesephases.iiiPrefaceA version of Chapter 2 was has been published in M. Rozali, D. Smyth, andE. Sorkin. Holographic Higgs Phases. JHEP, 08:118, 2012. In collaborationwith my supervisor, Dr. Rozali, I undertook all numerical and analytic computa-tions and checks, and produced all figures for the publication. Dr. Evgeny providedadvice and guidance regarding numeric approaches and methods. Writing was acollaborative process between authors.A versions of Chapter 3 has been published in M. Rozali, D. Smyth, E. Sorkin,and J. B. Stang. Holographic Stripes. Phys. Rev. Lett., 110(20):201603, 2013. Aversion of Chapter 4 has been published in M. Rozali, D. Smyth, E. Sorkin, andJ. B. Stang. Striped order in AdS/CFT correspondence. Phys. Rev., D87(12):126007,2013. I contributed analytic calculations, including contributions in the derivationof the metric, equations, boundary and gauge conditions, the extraction of the re-sults using the techniques of holographic normalization, and the understanding ofthe thermodynamics of the system. I also contributed in discussions and meetingsto help with the numerics.A version of chapter Chapter 5 was published in M. Rozali and D. Smyth.Fermi Liquids from D-Branes. JHEP, 05:129, 2014. In collaboration with mysupervisor, Dr. Rozali, I undertook all numerical and analytic calculations andproduced all figures presented in the publication. I was also also principally re-sponsible for writing the paper with edits and suggestions being provided by Dr.Rozali.A version of chapter Chapter 6 was published in M. Rangamani, M. Rozali,and D. Smyth. Spatial Modulation and Conductivities in Effective HolographicTheories. JHEP, 07:024, 2015. In collaboration with my supervisor, Dr. Rozali,ivI undertook all numerical and analytic calculations and produced all figures pre-sented in the publication. Dr. Rangamani provided important insights during theearly stage of the project and during the stages of interpreting the results and writ-ing the manuscript. With the exception of Section 3.1, which was largely writtenby Dr. Rangamani, I was principally responsible for writing the paper. Edits andsuggestions were provided during writing by Dr. Rozali and Dr. Rangamani.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivating Holographic Physics . . . . . . . . . . . . . . . . . . 11.2 AdS/CFT and Gauge-Gravity Correspondence . . . . . . . . . . . 31.3 The Challenges of Condensed Matter and the Promise of Holography 61.4 The Mechanics of Numerical Holography . . . . . . . . . . . . . 81.4.1 Formulating the Lagrangian . . . . . . . . . . . . . . . . 91.4.2 Formulating the equations of motion and boundary condi-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Solving the equations . . . . . . . . . . . . . . . . . . . . 121.4.4 Processing the solutions . . . . . . . . . . . . . . . . . . 141.5 Holographic Higgs Phases . . . . . . . . . . . . . . . . . . . . . 171.6 Striped Order in AdS/CFT . . . . . . . . . . . . . . . . . . . . . 18vi1.7 Fermi Liquids from D-branes . . . . . . . . . . . . . . . . . . . . 201.8 Spatial Modulation and Conductivities in Effective HolographicTheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Holographic Higgs Phases . . . . . . . . . . . . . . . . . . . . . . . . 262.1 Introduction and Conclusions . . . . . . . . . . . . . . . . . . . . 262.2 Characterization of Gauge Theory Phases . . . . . . . . . . . . . 282.3 Top Down Model . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Electric flux lines . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Magnetic flux lines . . . . . . . . . . . . . . . . . . . . . 312.4 Application to Holographic Superconductivity . . . . . . . . . . . 342.4.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Ansatz and boundary conditions . . . . . . . . . . . . . . 352.4.3 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.4 Bulk properties of the solutions . . . . . . . . . . . . . . 402.4.5 Boundary properties of the solutions . . . . . . . . . . . . 432.4.6 Dependence on parameters . . . . . . . . . . . . . . . . . 442.4.7 Electric screening . . . . . . . . . . . . . . . . . . . . . . 472.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Holographic Stripes . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . 513.2 The Holographic Setup . . . . . . . . . . . . . . . . . . . . . . . 533.3 The Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Striped Order in AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . 634.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . 634.2 Numerical Setup: Einstein-Maxwell-Axion Model . . . . . . . . . 674.2.1 The model and ansatz . . . . . . . . . . . . . . . . . . . . 674.2.2 The constraints . . . . . . . . . . . . . . . . . . . . . . . 694.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . 704.2.4 Parameters and algorithm . . . . . . . . . . . . . . . . . 754.3 The Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76vii4.3.1 Metric and fields . . . . . . . . . . . . . . . . . . . . . . 764.3.2 The geometry . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Thermodynamics at Finite Length . . . . . . . . . . . . . . . . . 854.4.1 The first law . . . . . . . . . . . . . . . . . . . . . . . . 854.4.2 Phase transitions . . . . . . . . . . . . . . . . . . . . . . 864.5 Thermodynamics for the Infinite System . . . . . . . . . . . . . . 935 Fermi Liquids from D-Branes . . . . . . . . . . . . . . . . . . . . . 955.1 Introduction and Outline . . . . . . . . . . . . . . . . . . . . . . 955.2 Setup: Equations and Boundary Conditions . . . . . . . . . . . . 975.2.1 Bosonic sector . . . . . . . . . . . . . . . . . . . . . . . 975.2.2 Fermionic sector . . . . . . . . . . . . . . . . . . . . . . 995.2.3 Parameters and limits . . . . . . . . . . . . . . . . . . . . 1025.3 Bulk Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . . . 1035.3.1 Iteration procedure . . . . . . . . . . . . . . . . . . . . . 1035.3.2 Solutions in the probe limit . . . . . . . . . . . . . . . . . 1055.3.3 Including backreaction . . . . . . . . . . . . . . . . . . . 1095.4 Boundary Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.1 Fermion density . . . . . . . . . . . . . . . . . . . . . . 1175.4.2 Retarded Green’s function . . . . . . . . . . . . . . . . . 1185.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 1196 Spatial Modulation and Conductivities in Effective Holographic The-ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.1 Introduction and Outline . . . . . . . . . . . . . . . . . . . . . . 1246.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2.2 Perturbations and linear response . . . . . . . . . . . . . 1336.3 Analytic Expressions for the DC Conductivities . . . . . . . . . . 1366.3.1 Response from hydrodynamic perspective . . . . . . . . . 1366.3.2 Membrane paradigm for response . . . . . . . . . . . . . 1386.4 Transport Results for Holographic Systems . . . . . . . . . . . . 1446.4.1 DC conductivities . . . . . . . . . . . . . . . . . . . . . . 144viii6.4.2 Metal-insulator transitions . . . . . . . . . . . . . . . . . 1496.4.3 Optical conductivity . . . . . . . . . . . . . . . . . . . . 1526.4.4 High temperature limit . . . . . . . . . . . . . . . . . . . 1536.5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . 163Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A Appendix: Holographic Higgs Phases . . . . . . . . . . . . . . . . . 175A.1 Vortex Solutions in Flat Spacetime . . . . . . . . . . . . . . . . . 175A.2 Details of the Numerics . . . . . . . . . . . . . . . . . . . . . . . 177B Appendix: Striped Order in AdS/CFT . . . . . . . . . . . . . . . . . 180B.1 Asymptotic Charges . . . . . . . . . . . . . . . . . . . . . . . . . 180B.1.1 Deriving the charges . . . . . . . . . . . . . . . . . . . . 180B.1.2 Explicit expressions for the charges . . . . . . . . . . . . 182B.1.3 Consistency of the first laws . . . . . . . . . . . . . . . . 184B.2 Further Details about the Numerics . . . . . . . . . . . . . . . . . 186B.2.1 The linearized analysis . . . . . . . . . . . . . . . . . . . 186B.2.2 The equations of motion . . . . . . . . . . . . . . . . . . 188B.2.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 191B.2.4 Generating the action density plot . . . . . . . . . . . . . 194B.2.5 Convergence and independence of numerical parameters . 195C Appendix: Fermi Liquids from D-branes . . . . . . . . . . . . . . . 202C.1 Charges and Stress Tensor Components . . . . . . . . . . . . . . 202C.2 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203C.2.1 Numerical techniques . . . . . . . . . . . . . . . . . . . . 203C.2.2 Convergence checks . . . . . . . . . . . . . . . . . . . . 204D Appendix: Spatial Modulation and Conductivities in Effective Holo-graphic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207D.1 Numerical Procedure and Implementation Details . . . . . . . . . 207D.2 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . 208D.3 Fit to Drude Form . . . . . . . . . . . . . . . . . . . . . . . . . . 223ixList of TablesTable B.1 The maximum critical temperatures and corresponding criticalwavenumbers for varying c1. . . . . . . . . . . . . . . . . . . 187Table B.2 Behavior of physical quantities with the cutoff for c1 = 8 andLµ/4 = 0.75 and for fixed grid resolution dρ,dx∼ 0.02. . . . 198Table B.3 Comparison of the constraint violation, measured by the schematicconstraint equation∑i hi, to the scale set by the individual terms,∑i |hi|, for grid size dρ,dx∼ 0.01. . . . . . . . . . . . . . . . 200Table D.1 Testing the fit to the Drude form of the conductivity and check-ing agreement between the homogeneous and inhomogeneoussolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230xList of FiguresFigure 2.1 Wilson loop stretched between widely separated sources on theboundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.2 World volume of D1 brane stretched between widely separatedmagnetic sources on the boundary . . . . . . . . . . . . . . . 33Figure 2.3 Profile of the vortex matter fields at ' 0.89Tc . . . . . . . . . 41Figure 2.4 The effect of the boundary action on the profile of the bulkfields with α = 3 . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.5 Comparison of the translationally invariant and vortex free en-ergy densities for ' 0.91Tc . . . . . . . . . . . . . . . . . . . 43Figure 2.6 Comparison of the bulk free energies of the vortex solutionsfor α = 0 and α = 3 . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.7 The boundary free energy density of the soliton for several val-ues of the temperature relative to the critical temperature, Tc . 45Figure 2.8 The profile of the boundary free energy density of the vortexas a function of α at temperature ' 0.91Tc . . . . . . . . . . . 46Figure 2.9 The total boundary free energy and the string tension (internalenergy) as a function of temperature below the critical temper-ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.10 Boundary free energy density as function of the boundary gaugecoupling α . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 2.11 Zero frequency two point function of the boundary gauge fieldin the limit of small momentum . . . . . . . . . . . . . . . . 50Figure 3.1 Metric function profiles for θ ' 0.11 and c1 = 4.5 . . . . . . 55xiFigure 3.2 Left panel: The variation along x of the size of the horizon inthe y direction. Right panel: Ricci scalar relative to that ofReissner-Nordstro¨m-AdS (RN) black hole at θ ' 0.003 overhalf the period . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 3.3 Difference in the thermodynamic potentials between the in-homogeneous phase and the RN solution for c1 = 8, plottedagainst the temperature . . . . . . . . . . . . . . . . . . . . . 59Figure 3.4 The entropy of the inhomogeneous solution and of the RN so-lution for c1 = 8 . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 3.5 A contour plot of the free energy density, relative to the ho-mogenous solution . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.1 A summary of the boundary conditions on the domain of oursolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.2 Metric function profiles for T/Tc ' 0.11 . . . . . . . . . . . . 77Figure 4.3 The profiles of At , relative to the corresponding RN solutions,and Ay and ψ for T/Tc ' 0.11 . . . . . . . . . . . . . . . . . 78Figure 4.4 Magnetic field lines for solution with T/Tc ' 0.07 . . . . . . 79Figure 4.5 Ricci scalar relative to that of RN black hole, R/RRN−1, RRN =−24, for T/Tc ' 0.054 over half the period of solution . . . . 80Figure 4.6 The embedding diagram of constant x spatial slices, as a func-tion of x at given y for T/Tc ' 0.035 . . . . . . . . . . . . . . 81Figure 4.7 Radial dependence of the normalized proper length along x forT/Tc ' 0.054 . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 4.8 Temperature dependence of the proper length of the horizonalong the stripe . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.9 The extent of the horizon in the transverse direction, ry, as afunction of x . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 4.10 The dependence of the size of the neck and the bulge on tem-perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 4.11 Behavior of rnecky /rbulgey at small temperatures. . . . . . . . . . 84Figure 4.12 The grand free energy relative to the RN solution for severalsolutions of different fixed lengths at c1 = 8 . . . . . . . . . . 87xiiFigure 4.13 The grand free energy relative to the RN solution for c1 = 4.5and fixed Lµ/4 = 2.08 . . . . . . . . . . . . . . . . . . . . . 88Figure 4.14 The observables in the grand canonical ensemble for c1 = 8and Lµ/4 = 1.21 plotted with the corresponding quantities forthe RN black hole . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.15 The difference in canonical free energy, at c1 = 8 and fixedLN/4= 1.25, between the striped solution and the RN black hole 91Figure 4.16 The entropy of the inhomogeneous solution and of the RN so-lution for c1 = 8 . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 4.17 Action density for c1 = 8 system relative to the RN solution . . 94Figure 5.1 Bulk field profiles resulting from varying µ with m0 = 1, β =−0.01 and mψ = 10 . . . . . . . . . . . . . . . . . . . . . . 106Figure 5.2 Bulk field profiles resulting from varying mψ with m0 = 1, µ =−15.7154, and β =−0.001 . . . . . . . . . . . . . . . . . . 107Figure 5.3 The bulk field profiles resulting from varying m0 with β =−0.001, µ =−15.7154 and mψ = 10 . . . . . . . . . . . . . 108Figure 5.4 The bulk field profiles resulting from varying β with m0 = 1,mψ = 10, µ =−15.7154 . . . . . . . . . . . . . . . . . . . . 109Figure 5.5 The backreacted, normalized bulk field profiles resulting fromvarying µ with β =−0.01, ε = 0.1, m0 = 1, mψ = 10 . . . . 110Figure 5.6 The backreacted, normalized bulk field profiles resulting fromvarying m0 with µ =−15.7154,ε = 0.1,β =−0.001,mψ = 10 112Figure 5.7 The bulk field profiles resulting from varying mψ with µ =−15.7154, m0 = 1, β =−0.001, ε = 0.1 . . . . . . . . . . . 113Figure 5.8 The bulk field profiles resulting from varying β , with µ =−15.7154, m0 = 1, mψ = 10, ε = 0.1 . . . . . . . . . . . . . 115Figure 5.9 The bulk field profiles resulting from varying ε with β =−0.001,m0 = 1, mψ = 10 and µ =−15.7154 . . . . . . . . . . . . . 116Figure 5.10 The change in volume near the embedding cap off as a functionof several paramters . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.11 The magnitude of the charge density in the dual QFT as a func-tion of various parameters . . . . . . . . . . . . . . . . . . . 122xiiiFigure 5.12 Green’s function signature in the (k,w) plane region near Fermisurface at k f = 15.3676 . . . . . . . . . . . . . . . . . . . . . 123Figure 6.1 Plots of the DC electrical and thermoelectric conductivities againsttemperature for various theories parameterized by υ , with C =1.5,k = 1 held fixed . . . . . . . . . . . . . . . . . . . . . . 146Figure 6.2 Plots of both forms of the thermal conductivity, κ and κ¯ as afunction of Tµ . . . . . . . . . . . . . . . . . . . . . . . . . . 147Figure 6.3 Plots of the Lorenz factors associated with L and L¯ as a func-tion of Tµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 6.4 The phase plot in the (υ ,k) plane illustrating the metal insula-tor transition . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 6.5 Plots of the background, bulk fields for T = 0.0015, k = 2.02and υ = 0 and υ = 0.002 . . . . . . . . . . . . . . . . . . . . 151Figure 6.6 The DC conductivity in the limit υ = 0 for different latticewavenumbers,k. . . . . . . . . . . . . . . . . . . . . . . . . . 152Figure 6.7 The effect of the parameter υ on the real and imaginary partsof the AC conductivity as Tµ is lowered . . . . . . . . . . . . . 154Figure 6.8 A continuation of the sequence of figures begun in Figure 6.7 155Figure 6.9 The real part of the AC conductivity as a function of ωT for avariety of Tµ and υ . . . . . . . . . . . . . . . . . . . . . . . 156Figure 6.10 A continuation of the sequence of figures begun in Figure 6.9 157Figure 6.11 The behaviour of the diagnostic function, F(w), as we scan formid-range scaling behaviour as a function of υ . . . . . . . . 158Figure 6.12 A continuation of the sequence of figures begun in Figure 6.11 159Figure 6.13 The behaviour of the diagnostic function F(w) as we scan formid-range scaling behaviour as a function of temperature . . . 160Figure 6.14 A continuation of the sequence of figures begun in Figure 6.13 161Figure 6.15 The Infrared (IR) to Ultraviolet (UV) scaling turning point forthe electric and thermoelectric conductivity . . . . . . . . . . 162Figure B.1 The critical temperatures at which the RN black brane becomesunstable, for varying axion coupling c1 . . . . . . . . . . . . 188xivFigure B.2 The data underlying Fig. 4.17 . . . . . . . . . . . . . . . . . 195Figure B.3 The behavior of the L2 norm of the residual during the relax-ation iterations for c1 = 8, T0 = 0.04 and Lµ/4 = 0.75 . . . . 196Figure B.4 The value of the scalar field condensate for varying grid sizesfor c1 = 8 and Lµ/4 = 0.75 . . . . . . . . . . . . . . . . . . 197Figure B.5 The bulk profiles of the weighted constraints for c1 = 8 andLµ/4 = 1.21 . . . . . . . . . . . . . . . . . . . . . . . . . . 199Figure C.1 The convergence with N for the embedding function, gaugefield, and charges . . . . . . . . . . . . . . . . . . . . . . . . 205Figure D.1 The convergence of the log of the norm of the difference of thesolutions as a function of Nz for temperatures of (0.0016,0.0021)for a variety of υ with Nx = 45 . . . . . . . . . . . . . . . . . 210Figure D.2 The convergence of the log of the norm of the difference ofthe solutions as a function of Nx for the same values of υ andtemperature as the previous plot and with with Nz = 45 . . . . 211Figure D.3 The convergence of the log of the norm of the gauge conditionas a function of Nz for a variety of υ . . . . . . . . . . . . . . 213Figure D.4 Convergence of the log of the norm of the gauge condition forparameters as in Figure D.3 . . . . . . . . . . . . . . . . . . 214Figure D.5 Convergence of the log of the norm of the residues with bothNz and Nx increasing but with Nx lagging Nz by 20 . . . . . . 215Figure D.6 Semi-logarithmic plots of the norm of the difference in solu-tions, the norm of horizon auxiliary conditions, and the normof the gauge conditions versus N at a low and high tempera-tures for wT = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . 216Figure D.7 A continuation of the sequence of figures begun in Figure D.6. 217Figure D.8 Plots of the real and imaginary conductivities versus N for thesame choice of parameters as in Figure D.6 and Figure D.7 . . 218Figure D.9 A continuation of the sequence of figures begun in Figure D.8 219xvFigure D.10 Semi-logarithmic plots of the norm of the difference in solu-tions, the norm of horizon auxiliary conditions, and the normof the gauge conditions versus N at a low and high tempera-tures for wT = 7 . . . . . . . . . . . . . . . . . . . . . . . . . 220Figure D.11 A continuation of the sequence of figures begun in Figure D.10. 221Figure D.12 Plots of the real and imaginary conductivities versus N for thesame choice of parameters as in Figure D.10 and Figure D.11 222Figure D.13 A continuation of the sequence of figures begun in Figure D.12 223Figure D.14 The auxiliary horizon constraints corresponding to Figure 6.11and Figure 6.12 . . . . . . . . . . . . . . . . . . . . . . . . . 224Figure D.15 A continuation of the sequence of figures begun in Figure D.14 225Figure D.16 The auxiliary horizon constraints corresponding to Figure 6.13and Figure 6.14 . . . . . . . . . . . . . . . . . . . . . . . . 226Figure D.17 A continuation of the sequence of figures begun in Figure D.16 227Figure D.18 The fits of the Drude parameters τ,K to the AC electric con-ductivities for two moderate values of Tµ for 0.01≥ υ ≤ 0.1 . 228Figure D.19 A continuation of the sequence of figures begun in Figure D.18 229xviGlossaryA list and explanation of a selection of contractions used throughout this thesis.QFT Quantum Field TheoryCFT Conformal Field Theory: A conformally invariant QFT.N=4 SYM N = 4 Supersymmetric Yang-Mills: A superconformal quantum fieldtheory in 4 dimensions with 4 superchargesADS Anti-deSitter Space: A vacuum solution to the Einstein equations of con-stant negative curvature in D dimensions. In all cases in this thesis, unlessotherwise specified, we will be interested in D = 4.ADS/CFT The first example of the gauge-gravity duality. A duality between AdS5×S5 and N = 4 SY M.UV Ultraviolet: The high energy behaviour of a quantum field theory as under-stood within the context of the Wilsonian picture of renormalization groupflow. When used in the context of the dual supergravity theory commonlyrefers to the asymptotically AdS region near the conformal boundary.IR Infrared: The low energy behaviour of a quantum field theory as understoodwithin the context of the Wilsonian picture of renormalization group flow.When used in the context of the dual supergravity theory commonly refersto the deep interior of the spacetime.PDE Partial Differential Equation: A differential equation involving partial deriva-tives of the dependent quantities by more then one independent variable.xviiODE Ordinary Differential Equation : A differential equation involving differenti-ation of the dependent quantities by a single independent variable.D-BRANES Dirichlet branes: Solitonic excitations within string theory localizedin p dimensions and with coupling proportional to the inverse of gs. Theyare the start and end points for open strings.P-BRANES The manifestation of D-branes within the classical theory of super-gravity.RG Renormalization Group: The flow of couplings and operators with energyscales as described by the Wilsonian view of renormalization.SOR Successive Over Relaxation: A robust but slow and numerically intensiveprocess for solving non-linear PDEs.DC Direct Current: In the dual QFT studying the DC transport properties involvesperturbing the theory by a constant external field orientated in one of thespatial directions and measuring the response.AC Alternating Current: As for the DC transport properties except the externalfield now has a frequency dependence.RN Reissner-Nordstro¨m-AdS: A charged black hole solution of Einstein-Maxwelltheory with asymptotically AdS geometry in D dimensions. In all contextsin which it is used in this thesis this refers to the planar solution found inPoincare´ coordinates with D = 4.xviiiAcknowledgmentsI would like to thank all of those who have helped me professionally during mytime as a graduate student at UBC. First and foremost this means my supervisor,Moshe Rozali, whose knowledge, guidance and judgement was essential in guid-ing my development as theoretical physicist and researcher. His consistent com-mitment to always being available for consultation regarding the physics, math-ematics and writing aspects of projects was an immeasurable source of help andsupport throughout my degree.I would like to thank my collaborators: Jared Stang, Evgeny Sorkin, and MukundRangamani. It was an my great fortune to be able to work with such knowledge-able, industrious and dedicated researchers. In addition I would like to thank Ro-man Baranowski from the Orcinus’ Westgrid cluster for helping me to both obtainand effectively utilize the computational resources I required.On a personal level I would also like to thank my office mates, Anton Smessaertand Amanda Parker, whose company and conversation made my tenancy in theHennings’ Building much more pleasant. I would like to thank my friends androommates, Peter Hurley, and, Andrew Chapman, whose friendship made my timeas a graduate student a far more enjoyable affair.Finally I would like to thank my parents and my sister for their continual loveand support, both during my graduate studies and during all the years beforehand.xixChapter 1IntroductionThis thesis aims to provide an organized collection of published research projectsin which I participated during the course of my graduate studies at UBC. It is myhope that this may be useful to future graduate students interested in the increas-ingly relevant and sophisticated applications of numerical methods to holographicsystems. I will start by describing the motivation behind his line of study from botha condensed matter and string theory perspective. I will then attempt to place eachof the research projects in context, both from the point of view of the applied stringtheory community and from the perspective of my own evolving research interestsas a graduate student in string theory.1.1 Motivating Holographic PhysicsThe holographic correspondence aims to map difficult problems of strongly cou-pled physics to more manageable gravitational problems in higher dimensions.Most conventional approaches to solving quantum mechanical problems involvemodifying (or perturbing) the energy function(al) of one of a handful of exactlysolvable problems. The new problem is then treated order by order in a perturbationexpansion which is controlled by a small, dimensionless parameter. If this parame-ter is not small but instead large the new theory cannot be treated as a modificationof the known solution. Such theories are known as strongly coupled theories andare difficult to study.1Holographic physics allows us to map such problems, via the tools of stringtheory, to theories of classical gravity in higher dimensions. In this formulationthe quantum mechanical description of the theory, with its inherent strong couplingdifficulties, has been replaced by a description in terms of classical Einstein gravity.The puzzle of how to calculate scattering rates and correlation functions without thebenefit of perturbation theory is replaced by the challenge of solving the Einsteinand matter equations of motions in a higher dimensional spacetime. While this isnot an easy problem, significantly more progress can be made than when workingwithin the quantum mechanical formulation of the theory.The greatest major obstacle in the modelling process is discovering how to mapthe physics we are interested in from the quantum formulation of the theory to thegravitational formulation. A common approach, known as bottom up holography,is to pick a gravity Lagrangian which we believe will correspond to the quantummechanical physics of interest. This is done in the understanding that the gravityLagrangian we choose will be included in many larger Lagrangians for which theexact mapping can be derived. This approach is also motivated by the desire tomodel aspects of quantum physics which are common to many strongly coupledquantum theories as opposed to any particular one.This thesis applies this approach to the study of condensed matter phenom-ena such as, spontaneous symmetry breaking, non-Fermi liquids, metal insulatorphase transitions and high temperature superconductors. We do this by derivingand solving the equations of motion in the classical gravity scenario, and from thesolutions, extracting information about the thermodynamic and transport propertiesof the quantum mechanical theory. This process entailed a mixture of analytic andnumeric work. The analytic components included the derivation and processingof the equations of motion, boundary conditions, and gauge conditions associatedwith classical supergravity. The numeric work involved solving these equationsfor a variety of parameter settings and processing the results to extract informationabout the dual quantum theory.In the rest of this introduction we will examine the evolution of numericalholography and the details of the modelling process in more depth. We will thengive a brief overview of each of the projects which will be described in detail inlater chapters, and provide some context of their place within the evolution of the2field.1.2 AdS/CFT and Gauge-Gravity CorrespondenceThe development of ADS/CFT correspondence, now almost two decades old, markedthe beginning of a rapid shift of research interests within the string theory commu-nity. This was one of the first explicit examples of a duality between a quantumfield theory and a higher dimensional theory of classical gravity. A good intro-duction to the initial steps of formulating this correspondence can be found in theclassic papers of [135] and [98]. In addition a very thorough review encompass-ing the initial stages of derivation, justification and exploration of this duality isprovided in [2]. Other excellent introductions to the correspondence may be foundin [102], [111], [70], [99] and [122]. Here we restrict ourselves to giving a briefoverview of how the original (ADS/CFT) correspondence was derived, followingclosely the arguments of [2]. We then, again very briefly, outline how this the-ory may be adapted to explore less symmetric theories including those that are ofinterest in condensed matter and particle physics applications.The original ADS/CFT correspondence stated that there is an exact correspon-dence between the apparently disparate theories of N = 4 Supersymmetric Yang-Mills (N=4 SYM) and AdS5× S5 at least in the limit where the number of colors,N , and the effective coupling, λ are large. This means that any physical observ-able in one theory has a corresponding quantity in the other. The fact that theduality is between a strongly coupled Quantum Field Theory (QFT) and weaklycoupled classical gravity in one higher dimension make it both difficult to proveand explore and a very useful tool. One side of the duality may always be exploredperturbatively however calculating, or even defining, the corresponding quantity inthe dual theory may be very difficult. However, given the large amount of bothcircumstantial and indirect evidence for the duality and the successful matching ofcalculations between both theories in cases where they can be made, it is generallyaccepted that the duality is valid. This is the perspective we will take throughoutthis thesis. One may then use the duality to explore strongly coupled physics whichis inaccessible to conventional perturbative approaches.We now reproduce one argument for the correspondence- following closely the3description given in [2]. The argument proceeds by comparing the behaviour of themassive, extended objects which exist within both string theory and classical grav-ity. These are the Dirichlet branes (D-BRANES) and (P-BRANES) respectively andmay be viewed as complementary representations of the same object. By exam-ining the behaviour of the two representations when a particular limit is taken theexistence of the duality may be inferred. Below we consider each representationseparately before combining our observations.P-BRANES interpretation: We search for classical black hole solutions formed by a stack ofN P-BRANESwith p = 3 embedded in flat 9+ 1 dimensional spacetime when N >> 1.These solutions may be constructed and are found to be valid when R>> ls.Here ls is the string length scale which is related to the Planck length aslp = g14s ls where gs is the string coupling1. R is the radius of curvature ofthe spacetime which, in the class of solutions in which we are interested,is found to be R4 ∝ gsl4sN . In addition we require lp < ls so that stringloop corrections are suppressed. These considerations together mean thatour supergravity black hole is a good description of the physics providedthat: gsN ∼ R4l4s >> 1. We now observe that, from the point of view of anobserver at infinity, there are two forms of low energy excitation for this sys-tem. These are: any excitation taken sufficiently far down the gravitationalwell associated with the near horizon geometry of the black hole horizon,and massless excitations in the bulk spacetime away from the gravity well.Furthermore we note that in the large N (large R) limit the two sectors de-couple from each other. In this limit near horizon excitations cannot climbup the well to reach an observer at infinity. Likewise it may be checked thatthe massless modes of the bulk do not interact with (or “scatter” from) theblack hole horizon geometry. Crucially it may also be checked that the nearhorizon geometry has the form of AdS5×S5.D-BRANES interpretation: We consider a stack of N D3-branes embedded in the same 9+ 1 dimen-sional spacetime where again we take N to be large. The energy scalefor this system is set by 1/ls. We are interested in the low energy, long1We remind the reader that the sting coupling is not a fixed constant but is set dynamically by thetheory as a function of the dilaton configuration.4distance limit realized by ls → 0 with all other dimensionless parametersheld fixed. We therefore integrate out all massive modes to acquire a non-renormalizable, effective action of the form S = Sbulk + Sbrane + Sint . HereSbulk, Sbrane and Sint are, respectively, the action contributions coming fromthe closed string modes in the bulk, the open string modes stretching betweenthe branes, and the interaction between these two sectors. It can be checkedthat Sint is proportional to positive powers of ls and therefore vanishes in thislimit. We are therefore left with two decoupled low energy actions. Thesetake the form of N=4 SYM on the brane worldvolume and a classical the-ory of gravity in 9+ 1 dimensions. This approximation is valid in the limitwhere gsN ∼ g2Y MN ∼ R4l4s<< 12. The first two parts of the inequality areequivelant statements that the open string interactions are suppressed andthe N=4 SYM interpretation remains valid. The last part of the inequality isa combination of the gsN << 1 and ls→ 0 statements applied to the radiusof curvature object, R, which was defined during the p-brane analysis above.Putting things together we see that, by interpreting the large N limit in two dif-ferent ways, we obtain the same theory of low energy bulk supergravity togetherwith either i) AdS5× S5 classical gravity or ii) N=4 SYM at strong coupling. It isnatural to associate the the two interpretations with each other. This interpretationis borne out by many other similarities and the matching of quantities which maybe calculated in both theories. For details of these explorations we refer the readerto the references mentioned above.The prototypical duality described above has since developed into the fieldof study known as the gauge-gravity duality. This research program explores theduality between strongly coupled QFTs with an Ultraviolet (UV) conformal fixedpoint and asymptotically Anti-deSitter Space (ADS) spacetimes. This generaliza-tion upon the ADS/CFT correspondence is necessary as most QFTs in which we areinterested in the context of particle or condensed matter physics do not possess thesame high degrees of symmetry as the prototypical N=4 SYM. In condensed matterapplications, such as those discussed in this thesis, we are interested in exploring2In writing the second part of the equality we have used the fact that the string coupling may berelated to the Yang-Mills coupling of the low energy effective theory as gs ∝ g2Y M5emergent, strongly coupled behaviour in the Infrared (IR) of the QFT. To model thisbehaviour we deform the UV fixed point by relevant operators which will cause thetheory to flow to some non-trivial IR fixed point. From the bulk spacetime point ofview, where the radial coordinate corresponds to the QFT energy scale, this corre-sponds to Cauchy evolution from the asymptotically ADS spacetime near the con-formal boundary to a non-trivial spacetime configuration deep in the bulk. Thisspacetime encodes the thermodynamic and transport properties of the new con-densed matter phase. Constructing these spacetimes and extracting and analysingtheir properties for a variety of condensed matter phases will be the main goal ofthis thesis.1.3 The Challenges of Condensed Matter and thePromise of HolographyGiven string theory’s long-standing close association with particle physics, the con-struction of holographic models of particle physics was amongst the first uses ofthe gauge-gravity duality. It was not until the end of the first decade of develop-ment that the increasing sophistication of the tools available and a greater aware-ness amongst the high energy community of the types of strongly coupled fieldtheory phenomena which may be created in a lab setting led to a greater focus onexploring condensed matter phenomena within a holographic setting.This new research direction offered both potential of insights into long standingtheoretical challenges in condensed matter and, from a condensed matter physicistsperspective, an additional tool to aid in the construction of other theoretical phasesof matter. Excellent lectures on the initial progress made in understanding the ther-modynamic and transport properties of holographic superconductors, (non-)Fermiliquids, and quantum critical phases can be found in [54], [65], and [92]. Theseefforts were successful in reproducing many of the qualitative features of thesecondensed matter systems; for example the order of the phase transition and valuesof the critical exponents of the superconductivity transition. The fact that qualita-tive similarity was achieved with real world condensed matter systems is indicativeof a general trend within holographic condensed matter systems. The nature of themodelling process suggests that the qualitative results obtained are those which are6most likely to be generic to a class of quantum field theories. As such, these holo-graphic condensed matter models can be seen as representative of a broad class offield theories displaying certain strongly coupled quantum properties.In order to gain some intuition into this universal 3 nature of holographic con-densed matter models it is helpful to briefly consider the mechanics of the mod-elling process. An applied string theorist attempting to investigate some aspects ofa strongly coupled condensed matter system must find the bulk spacetime whichencodes these features, via the mechanics of the gauge-gravity duality, in its mat-ter and metric fields. One could attempt to generalize the process by which theADS/CFT correspondence was motivated earlier in this introduction and constructthe low energy physics as a consistent reduction of a known string theory. Sucha process would, however, be very difficult and liable to yield fields whose pres-ence and configuration was incidental to the physics under exploration. Thereforeit is an accepted practice to use only the minimum number of bulk componentsnecessary to capture the physics under consideration. This is done with the un-derstanding that these theories, together with various ancillary components, mightbe embedded within a suitable string theory reduction. From the point of view ofthe dual Conformal Field Theory (CFT) this approach corresponds to deformingthe UV of the theory by operators which are both relevant (in the Wilsonian sense)and important to the macroscopic physics we wish to investigate. The hope isthat the resulting IR physics should be representative of aspects of the qualitativebehaviour of many quantum field theories formed by the inclusion of additionalrelevant deformations.Given the initial success achieved in using holographic methods to explore thephysics of homogeneous condensed matter systems, a natural next step was to con-sider systems with reduced degrees of symmetry. In particular, one may considerthe breaking of translational invariance. Such non-translationally invariant sys-tems are widespread in the real world. Examples include diverse behaviour fromelectron-lattice interactions to the spontaneous breaking of translational invariancevia the formation of charge or spin density waves. Unfortunately, describing thesesystems from a holographic perspective is significantly more challenging than ho-3The use of the word universal here is intentional and intended to draw attention to the parallelswith renormalisation group flow.7mogeneous systems. The bulk equations governing the spacetime configuration,and therefore encoding the information about the dual strongly coupled QFT, arenow PDEs, as opposed to ODEs. While much progress had been made in extract-ing information from the ODEs via analytic and numerical methods, the solving ofPDEs requires a much greater degree of numerical and computational sophistica-tion. This thesis is written in the context of the growing usage of these techniqueswithin the string theory community. In the next section I provide an overview ofsome of the considerations and steps common to many applied numerical hologra-phy projects. I then provide a brief discussion of the motivation, goals and contextbehind each of the research projects in which I participated before examining themin detail in the body of this thesis.1.4 The Mechanics of Numerical HolographyThe use of sophisticated numerical methods within string theory has been confined,with several notable exceptions (see for example [131] and the author thereof), tothe last decade. The utilization of these techniques to solve otherwise intractablePDEs has drawn heavily on the wide body of existing knowledge within the numeri-cal general relativity community. As will hopefully be demonstrated in the body ofthis thesis numerical approaches can form a very useful tool for extracting informa-tion about thermodynamic and transport properties of strongly coupled QFT withreduced degrees of symmetry. It should however be noted that experience, bothpersonal and of the community as a whole, has shown that accurately modellingthe sophisticated mathematical machinery of gauge-gravity duality using numer-ically techniques is a difficult task. The removal of analytic control compoundsthe long standing lack of any experimental probes of string theory phenomena tothe point that great numerical care must be taken in order to avoid misleading orincorrect conclusions.Ultimately the role of numerical methods in holography, much like the roleof holography in condensed matter physics, is in the opinion of the author notyet fully decided. Whether these approaches will prove fruitful in providing thecrucial insights needed to understand strongly coupled phases of condensed matterremains to be determined. With this in mind we proceed to give a brief overview8of the procedure for formulating and solving a problem in numerical, holographiccondensed matter problem.1.4.1 Formulating the LagrangianThe applied string theorist attempts to capture strongly coupled physics of QFT via“geometrizing” it within an asymptotically ADS spacetime. To find this dual space-time from first principals would require repeating the decoupling limit procedureby which the ADS/CFT was initially discovered, and which is described in detail in[2]. Compared to this prototypical case which is formulated for the highly symmet-ric N=4 SYM theory, the QFT will generically have a variety of additional fields,couplings and broken symmetries which are relevant for describing the stronglycoupled physics of interest. While this form of derivation is possible in a limitedset of cases it is often preferable to simply conjecture a Lagrangian which one canargue must be embedded within some string theory. This approach is motivatednot only by a desire to reduce calculational complexity but also from the intuitionthat we wish to study strongly coupled physics generic to a large class of QFT. Thesupergravity duals of these QFTs should therefore have universal components intheir Lagrangians which encapsulate this physics. Indeed given the current preci-sion of holographic techniques it is essentially only such universal physics whichwe can reliably probe. This approach is known as “bottom-up” or “applied” holog-raphy and has had considerable success in modelling aspects of condensed matterphysics such as topological insulators [74, 84], superconductivity [31, 59, 103] ,non-Fermi liquids [5, 47, 119], inhomogeneous phases [11, 34, 73, 87], and metal-insulator transitions [43].A commonly used alternative approach to constructing a new bulk Lagrangianfrom scratch is to work in a well understood holographic model— for example theoriginal ADS/CFT model— and to introduce additional stringy degrees of freedom,known as D-BRANES, to construct phenomenological models of the physics ofinterest. These objects are dynamical soliton solutions within string theory whichact as end points for extended strings. They are useful in this context as in theclassical supergravity limit the D-BRANES may be introduced as “probe branes”which do not backreact on the geometry. The low energy modes of the strings9stretching between the D-BRANES introduce additional matter field content intothe theory. Therefore one can, by choosing an appropriate number and a stablearrangement of D-BRANES, effect precision control over the fields and couplingsappearing in the Lagrangian. D-BRANES will be used extensively in two chaptersof this thesis— see Chapter 2 and Chapter 5.1.4.2 Formulating the equations of motion and boundary conditionsOnce one has specified the Lagrangian one may take the appropriate variationalderivatives to derive the equations of motion. If the Lagrangian is for the bulkspacetime then these will consist of Einstein and matter field equations. If onehas utilized probe D-BRANES to construct the Lagrangian the Einstein equationswill be replaced by equations describing D-brane embeddings in the bulk space-time. In all projects considered in this thesis we will be interested in finding staticor stationary solutions to these equations of motion. This corresponds to study-ing a dual QFT in equilibrium4. If, in addition, the QFT is spatially homogenousthen the equations reduce to coupled, non-linear ODEs. Generically, however, fornon-translationally invariant systems the equations will be PDEs5. In order forthese PDEs to have unique solutions, and therefore to be numerically solvable, onemust specify appropriate boundary conditions. These must be imposed in the deepinterior of space-time, at its asymptotic boundary and in the transverse spatial di-rections.The choice of boundary conditions is important as it is these that encode thestate we wish to probe in the dual QFT. This may be understood as follows: theradial direction of the space-time encapsulates the renormalisation group flow ofthe dual QFT. The degrees of freedom near the space-time boundary correspond tothe UV degrees of freedom in the QFT and the boundary conditions imposed therecorrespond to the form of the relevant deformations introduced into the CFT whichexists at the UV critical point. Similarly the IR behaviour of the QFT is capturedby the behaviour of metric and matter fields in the deep interior of the space-time.4See [76],[22] for examples of non-equilibrium, time dependent holographic constructions.5As will be discussed in Chapter 6, a highly profitable research program exists for exploring trans-lational invariance while maintaining the ODEs nature of the equations of motion; see for example[33, 35–37, 41].10The fact that the solutions of the bulk equations of motion take into account allradial positions is an intuitive way of grasping how the classical gravity solutionsgive a non-perturbative description of the strongly coupled QFT physics. It is inprincipal, and sometimes in practice [48], possible to construct the IR bulk space-time via inward radial evolution from the boundary in a geometrical equivalent ofRenormalization Group (RG) flow. Frequently however it is mechanically neces-sary to impose some smoothness or regularity conditions in the IR which encodesome knowledge or ansatz regarding the form of the IR physics in which we areinterested.This intuition regarding RG flow from the boundary to the interior of the space-time does however provide us with a guiding principle in the formulation of bound-ary conditions. In the UV all metric and matter fields decouple and the equationsof motion linearize. The fields then obey falloff conditions which depend only onthe dimensionality of the space-time and the mass and spin of the matter fields.The dual QFT statement is that we are near the conformally invariant critical pointwhere field operator scaling is simple and governed by critical exponents. The im-position of boundary conditions in the interior is more subtle and is governed bythe type of macroscopic physics we wish to study in the QFT. If one wishes towork at a finite temperature the interior boundary becomes a black hole horizon.This is a sensible thing to do given that, unlike in flat space, it is possible to havestable/eternal, positive-specific-heat, black hole solutions in AdS [63]. The radialposition of the horizon then has the interpretation in the dual QFT as the energyscale below which quantum effects are masked by thermal fluctuations. Differentforms of metric and matter functions in the near horizon limit correspond to dif-ferent classes of IR behaviour in the QFT. The art of specifying the appropriateansatz in the IR in order to find the behaviour of interest in the QFT is one of themost challenging aspects of any project. If one is working at zero temperatureand on the Poincare´ patch (or with an extremal black hole) a similar situation willapply- one must impose self consistent boundary conditions on the fields that in-corporate the IR physics. However finding zero temperature holographic models ofcondensed matter systems is a difficult problem and all projects discussed in thisthesis deal with systems at finite temperature. The numerical difficulties associatedwith examining the very low temperature behaviour of holographic models will be11discussed in Chapter 4 and Chapter 6 and the appendices thereof.1.4.3 Solving the equationsOnce one has equations and boundary conditions one can proceed to find a solution.As the equations in question are non-linear ODEs or PDEs the solution technique isinvariably numeric. Various options are available depending on the nature of theproblem:Shooting methods: Useful for solving problems involving nonlinear ODEs provided the numberof free parameters is small. This method utilizes the powerful integrationroutines available for initial value problems to integrate the equations fromthe IR to the UV6. Under the supervision of a root-finding algorithm, freeparameters in the IR7, such as the gradient of the fields, are adjusted and theintegration iterated until the UV boundary conditions are satisfied to within aprescribed tolerance. A technical complication results from the singular be-haviour of some terms in the equations near any black hole horizon in the IRand the conformal boundary in the UV. This singular behaviour results fromthe common choice of coordinate system near the horizon, and the diver-gence of the volume element near the boundary, respectively. It is thereforenecessary to solve the equations locally via power series expansion in the IRand UV. Boundary conditions may be set and solution information extractedvia appropriate matching of the region of integration onto these local solu-tion patches. This method is useful due to its ease of implementation- thereare high quality, inbuilt initial value routines in Mathematica for example.It is however ’unnatural’ in the sense that it treats a boundary value prob-lem as an initial value problem and relies on the efficiency of initial valueintegration to compensate for the amount of repetition involved. In the casewhere multiple shooting parameters are required the use of spectral methods[17, 126] is recommended as an alternative.Finite Difference: A conceptually simple and robust approach for solving PDEs. A rectangularmesh of grid points is defined over the domain of solution. The equations of6It turns out attempting to integrate in the opposite direction tends to be numerically unstable.7These free parameters are known as shooting parameters.12motion and boundary conditions are then written as finite difference equa-tions with values defined on these grid points. This is done via approximat-ing derivatives via differentials using a set of neighbouring points known asstencils (see [1] for an explicit forms of the different types of stencils to vari-ous orders of approximation). The solution of the system of finite differenceequations should then be the solution of the system of PDEs up to correctionsgiven by the accuracy of the discretization process. Therefore the numericsolution should converge to the true solution as the grid size goes to zero. Forthe non-linear systems of equations described in this thesis the solution can-not be found in one step via simple matrix inversion. Instead the equationsmust be linearized and solved iteratively via methods such as SuccessiveOver Relaxation (SOR) [109]. In these methods a series of “sweeps” overthe grid is carried out with the values of the fields at each grid point beingupdated based on those of their neighbours until convergence is obtained.While simple, this method has a number of disadvantages which mean that itis generally worth investing in more sophisticated methods. These include:• Even in ideal conditions the convergence to the continuum as a functionof the number of grid points is not particularly fast- it is a power lawdependent on the number of points used in the discretization of thederivatives (for a basic three point stencil the convergence is of orderN2). The addition of nonlinearities and/or additional constraints mayreduce this convergence rate significantly.• Since the basic stencils of the discretization are local objects they arenot particularly well suited to accurately capturing behaviours like asymp-totic falloffs. This is of particular relevance in our applications. In or-der to compensate for this one may have to go to quite large N whichplaces corresponding demands on computational memory.• Iterative methods such as SOR converge slowly for large systems ofnon-linear equations, particularly when the number of grid-points istaken to be large and additional complications such as constraint equa-tions are introduced.All of the above considerations were relevant in the work done in Chapter 2,13Chapter 3 and Chapter 4 where these issues arose. Reasons to use finite dif-ference approaches include non-analytic or very rapidly changing behaviourin the solutions which is not well captured by more global methods, such asspectral methods.Spectral methods: A very efficient technique for solving non-linear equations provided the so-lutions can be effectively represented by polynomials and are not too rapidlychanging. Instead of discretizing based on grid points one expands the equa-tions in a polynomial basis. In this thesis we have used the popular Cheby-shev polynomials. It can be shown that knowledge of the solution up toorder N in the Chebyshev expansion is equivalent to knowing the value ofthe solution at the first N+1 Gauss-Chebyshev points known as nodal points(the roots of the Chebyshev polynomials). In our implementations it is thevalue of the solution at these nodal points that is solved for. This method hasthe advantage that it convergence to the continuum solution is exponentialin N for linear equations. Non-linear equations can be handled via a simplegeneralization of Newtonian or quasi-Newtonian methods which convergemuch more quickly than methods like SOR. In addition it can be shown thatfor any given N the Chebyshev polynomials give the best resolution near theboundary for a finite domain [17, 126]. This is ideal for the purposes of ex-tracting information from the bulk space-time solutions. Spectral methodsmay also be used with a Fourier basis. This approach is useful for discretiz-ing periodic domains such as the transverse/ spatial directions of many of themodels discussed in this thesis Chapter 6.1.4.4 Processing the solutionsOnce solutions have been found, information regarding the state of the QFT may beextracted. We now describe in more detail how quantities are translated betweenthe bulk and the dual QFT. From the point of view of the QFT the imposition ofvalues on coefficients of the leading asymptotic falloffs as boundary conditionscorresponds to turning on sources for relevant operators with a particular ampli-tude. In effect we add variable/conjugate variable pairs to the QFT path integralin order to specify the state of the theory. These coefficients encode our control14parameters, such as temperature or chemical potential, which can be used to pa-rameterize the spacetime solutions8. Once we have solved the bulk equations ofmotion we have the required information to extract information about the expecta-tion values of operators in the dual QFT. To do so, we make use of a key feature ofthe correspondence which states that the generating function in the QFT is dual tothe Euclidean, on-shell bulk action. We therefore vary this onshell bulk action withrespect to the leading falloffs of the fields and take the limit where by the resultingquantity approaches the spacetime conformal boundary [10, 135]. Unfortunatelythe resultant quantity is in many cases divergent. In order to extract finite answerswe must employ the methods of holographic renormalisation [107, 108] to regulateand renormalize these quantities via addition of appropriate boundary terms to thebulk action. The end result is usually that the expectation value is given by thesubleading asymptotic falloff of the field in question9.One can also extract information about the QFT Green’s functions from thebulk solution. Generally, we are interested in the retarded Green’s function as thischaracterizes the response of the system to an external probe. As we are interestedin the response of the system to infinitesimal perturbations the problem is one oflinear response theory in the QFT and therefore the Green’s function is the ratio ofthe response to the source of a perturbation. Holographically this involves solvingthe linearized bulk equations around a known background solution. The lecturesin [65] offer a particularly good pedagogical treatment of this topic. Mechanically,we introduce a frequency modulated perturbation of the background and solve thelinearized equations which result. The Green’s function may then be extracted viacomparing the source and response, as given by the leading and subleading falloffsrespectively as described in [59, 65, 81]. In the case where the original bulk sys-tem is described by ODEs the linearization about the on-shell solution results in8It is almost always more convenient to obtain some physical parameters, such as entropy andtemperature, from the horizon geometry of the black hole.9There is the additional, logically distinct though operationally similar, restriction of ensuringthe consistency of the variational problem which from which the bulk equations of motion are de-rived. The criteria may be expressed as the conversation at the conformal boundary of an appropriatesymplectic norm constructed from the bulk fields [90]. If this is not the case then one is required toadd additional boundary terms to the action to ensure that the variational problem is well defined.Perhaps the most well know of such terms is the Gibbons-Hawking boundary term which must beadded to the Einstein Hilbert action when varying on space times with boundaries.15linear ODEs which may be solved via shooting or spectral methods as described inSection 1.4.3. If the behaviour of the bulk system is governed by PDEs, the solu-tion of the linearized PDEs around a numerical background can be a challengingendeavour— see Chapter 6.Another key feature of the solutions which we may wish to examine is theirthermodynamic properties. This is particularly relevant in Chapter 3 and Chapter 4of this thesis where we are interested in examining whether the inhomogeneousor homogeneous solutions are thermodynamically dominant within a variety ofthermodynamic ensembles. Additionally in cases where we find a phase traditionwe wish to determine the scaling of thermodynamic quantities, such as the freeenergy, near the critical point in order to determine the order of the transition- seealso Chapter 2. Thermodynamic quantities may be computed as follows:• The free energy of the QFT is known to be equal to the value of the Eu-clidean on-shell bulk action. Again this quantity will require normalizationvia appropriate boundary terms in the bulk action.• In the case where a black hole is present the surface of the horizon is knownto be a measure of the entropy of the system. In addition the Hawking tem-perature of the black hole of the system is equal to the temperature of thesystem in thermal equilibrium.• The asymptotic value of the time component of the gauge field dictates thechemical potential of the system10. Likewise the charge density is givenby the subleading falloff of the time component of the gauge field. This isin accordance with the argument presented above relating the leading andsubleading components of the asymptotic falloff of a bulk field with the con-jugate thermodynamic variable in the dual QFT path integral.• The expectation values the components of the stress energy tensor of the QFTmay be found via the appropriate falloffs of the corresponding componentsof the bulk metric. The extraction of these expectation values is most easily10This of course implicitly implies we have chosen a gauge where the time component of thegauge field, A0, is finite. This is the case for all projects described in this thesis.16derived and understood in Fefferman-Graham coordinates as described in[107, 108].1.5 Holographic Higgs PhasesIn Chapter 2 of this thesis we study the characterization of the Higg’s mechanismfrom the point of view of bottom up holography. The project motivation was toprovide a diagnostic for spontaneous symmetry breaking when the microscopicdegrees of freedom of the QFT are not well known. The first step was to understandhow the phases of a gauge theory may be characterized in a gauge invariant way.This is done via the Wilson loopW (C) = Tr(ei∫C A) (1.5.1)and its electromagnetic dual, the ’tHooft loop, [67, 68]. Whether these quantitiesscale as the area or perimeter of the loop as its perimeter increases characterizesthe phase of a gauge theory:• Confinement: W (C)∼ e−A(C) and T (C)∼ e−L(C)• Higgs Phase: W (C)∼ e−L(C) and T (C)∼ e−A(C)A gauge invariant diagnostic was necessary as gauge dependent quantities are noteasily translated across the duality. Holographically, the Wilson and ’tHooft loopsare computed via calculating the minimum surfaces of D(or F)-branes with bound-ary conditions prescribed by the loop in the QFT [97]. It emerges that holographi-cally the Higgs mechanism may be realized via the the formation of a vortex fromthe dissolution of a D1-brane within the worldvolume of D3-brane.This led us to propose that a suitable diagnostic of the Higgs mechanism inbottom-up models is the existence of vortex solutions for the bulk matter fields. Insupport of this idea, numerical vortex solutions were constructed within the holo-graphic model of superconductivity of [59] and building on the work of [87, 103].An additional complication was the requirement that these vortices be constructedin theories where the dual QFT has a genuine, dynamic gauge field. 11 This required11Dynamic in this context refers to the presence of an appropriate kinetic term in the action. The17the use of the techniques of [31, 100], which allow us to implement alternativeboundary conditions for the gauge field via the inclusion of additional boundaryterms in the bulk action. The result of these additions is the the formation of anemergent, weakly coupled sector within the strongly coupled QFT and a direct re-lation between the bulk and QFT gauge fields. Thermodynamic quantities werealso explored. This yielded some intriguing results such as the fact that the freeenergy of the vortex solutions continues to obey weakly coupled Landau-Ginzburgtype behaviour despite the fact that these vortices are embedded within a stronglycoupled gauge theory.This work was significant from my personal academic development as it markedmy first exposure to solving PDEs in the context of the gauge-gravity duality andusing the techniques of the correspondence to extract thermodynamic and transportproperties from these numeric solutions. The lessons learned in terms of analyticand numerical technology and best practice, and the insight gained into the holo-graphic dictionary were important preparation for more extensive projects under-taken later.From the perspective of the applied string community this work was signifi-cant as it both demonstrated the existence of, and explored the properties of, vortexsolutions within an emergent, weakly coupled sector of a strongly coupled gaugetheory. In addition it provided a mechanism for the identification of Higgs mech-anism in spacetimes where the microscopic details of the theory are not known: ifone can construct a bulk vortex solution for a range of control parameters the dualQFT is in the Higgs phase for that range of parameters.1.6 Striped Order in AdS/CFTChapter 3 and Chapter 4 of this thesis stem from work done as part of a collabo-ration studying the formation of holographic striped phases. This was motivatedby the linear perturbation calculations of [34] which showed that the inclusion ofan axion term in the action meant that the Reissner-Nordstro¨m black hole is dy-namically unstable to the formation of spatially modulated phases below a critical”genuine” refers to the requirement that the relevant symmetry group indeed be a gauge redundancyof the QFT.18temperature. Our goal was to construct a full, non-linearly backreacted solutionfor this system with an ansatz which would break translational invariance and al-low the formation of inhomogeneous phases in one of the QFT spatial directions.From a condensed matter perspective the ability to study such strongly coupledinhomogeneous phases would be significant as the emergence of quasi-one dimen-sional structures is thought to be important in the onset of superconductivity withinsome cuprate superconductors [19]. The system to be solved was complicated- in-volving 7 dynamical and 2 constraint equations. The identification of appropriatemetric ansatz, the implementation of the constraints, and the appropriate choice ofcoordinates and boundary conditions in the UV and IR regions posed significanttheoretical and computational challenges. In the end however solutions were ob-tained and their thermodynamic properties probed. Some of the key results foundare:• The existence of striped phases which spontaneously break translational in-variance and in which the stripes have momentum, electric current and mod-ulations in charge and mass density in the boundary theory. In the bulk amodulated horizon was seen be be present with interesting curvature effectsobserved. These include the growth of the volume element in the near hori-zon region and the fact that the curvature was seen to be greatest at the bulgein the horizon. Naively one might expect the curvature to be maximized atthe horizon neck however the planar extent of the horizon in the directionswhere symmetry is not broken mean that this is not the case.• A second order phase transition to striped phases in all ensembles (micro-canonical, canonical and grand canonical) below the critical temperature.This was found to be true for both the finite length and infinite system. Thedistinction between finite and infinite systems is significant as it correspondsto fixing either the periodicity or the length in ones choice of ensemble.Agreement between the three ensembles was important as it agrees with thegeneral theory of phase transitions as viewed from different ensembles. Inaddition it was seen that violation of this agreement could lead to possibleviolations of cosmic censorship 12.12A striped solution which is thermodynamically preferred by the grand canonical and canonical19• The entropy of the system was seen to tend to a finite value as the temper-ature was taken towards zero, in apparent contradiction with the results of[132, 133]. This result suggests that either entropy is shed by the systemvery quickly at a very low temperature, or an additional low temperatureinstability may be present which would change the horizon geometry andremove the residual entropy. It is possible that such a solution, if it exists,might require the breaking of additional symmetries.This project was significant as it provided the opportunity to work as part ofa multidisciplinary team on complicated numerical and analytic problem. Thelessons learned in terms of the gauge and constraint structure of Einstein equationsin ADS, numerical methods of solving PDEs, and thermodynamics in the contextof holography and black hole were all important lessons which would be carriedforward to future projects.From the perspective of the field, this project was important as it providedthe first full, cohomogeneity 2 solution with spontaneous symmetry breaking inADS. It proceeded to provide a thorough exploration of the thermodynamics of thesystem- discussing its behaviour in 6 ensembles and demonstrating the existence ofsecond order phase transitions. As such it represented an important step in the go-ing research program to understand the spontaneous breaking of global symmetriesof strongly coupled systems via the gauge-gravity correspondence.1.7 Fermi Liquids from D-branesChapter 5 involves the holographic investigation of Fermi liquids with the goalof better understanding holographic mechanisms to generate non-Fermi liquid be-haviour. These substances are of particular interest in condensed matter physicsfor their unusual thermodynamic and transport properties which violate expectedLandau Fermi liquid behaviour. This anomalous behaviour results from a lack ofa weakly interacting, quasiparticle description of the excitations of the Fermi sur-face as postulated by Landau theory. In addition, understanding non-Fermi liquidphases may provide clues to the origin of the high temperature superconductorensembles and yet dynamically unstable (not preferred microcanonically) cannot be the end point ofthe evolution. and would suggest discontinuous behaviour in the horizon topology.20instabilities to which they are known to be susceptible [19]. Previous work onthis topic produced non-Fermi liquid behaviour by coupling the Dirac action toEinstein-Maxwell action in the probe limit, such that the fermions did not back-react on the spacetime. The solution was found to be black hole with non-trivialfermonic hair [47]. In these solutions, the fact that proper length diverged as theblack hole horizon is approached served to act as a means of introducing largeamounts of dissipation into the system. Such an effect mirrors many toy modelsof condensed matter physics where non-Fermi liquid behaviour is realized by cou-pling the fermions to a very large number of gapless bosonic excitations. From abulk perspective these bosonic modes correspond to the metric fluctuations in thedeep IR, in the near horizon region of the black hole. This bosonic sector acts as adissipative bath and serves to give the quasi-particles a very short lifetime. How-ever this solution was found to be pathological as together with the ”genuine” fermisurface of interest it was found to contain O(N) Fermi surfaces in the deep IR, onefor every species of gluon present in the dual QFT. When filled in the Thomas-Fermi approximation and projected onto the boundary this lead to a continuum ofFermi surfaces in the QFT.In order to try and move beyond the Thomas-Fermi approximation it was de-cided to follow up on the work of [5, 6, 119] while taking additional inspirationfrom results on Minkowski brane embeddings in [50, 88, 101]. The idea was toconstruct backreacted solutions for a quantum bulk fermi liquid on the worldvol-umes of D-BRANES. Such a solution could form a suitable starting point fromwhich the formation of a non-Fermi liquid could emerge naturally, as functionof parameters. In order to avoid the previous pathologies it was important thatthe non-Fermi liquid behaviour emerge naturally from IR interactions between thefermionic and bosonic degrees of freedom provided by the D-brane embeddingfunction. The choice to work on a probe D-brane was made in an effort to avoidthe significant complications associated with the full gravity system as investigatedin [5]. The numerics of this project involved the solution of integro-differentialequations as the fermionic charge density couples to the bosonic embedding in anon-local fashion. The derivation of an iterative numerical scheme to successfullysolve these equations for a range of densities was non-trivial and required signifi-cant analytic and numeric experimentation. Some of the key of the solutions that21were found were:• Successful construction of fully backreacting, finite density fermionic solu-tions on the worldvolume of a probe D-brane in the Minkowski embedding.These geometries are supported against collapse by the Fermi pressure asso-ciated with a finite fermionic charge density.• The dual QFT is seen to possess O(1) Fermi surfaces and therefore thesesolutions overcome the difficulties associated with the Reissner-Nordstro¨m-AdS (RN) based constructions of [47, 92].• In the spirit of bottom up holography, a study was undertaken of the ther-modynamic and transport properties of this phase of matter as a function ofthe various control parameters in the theory. This confirmed the presence, inthe dual QFT of a gapped, compressive Fermi liquid with completely stablequasi-particle excitations in the vicinity of a finite number of Fermi surfaces.• It was found to signal that for the system to exhibit non-Fermi liquid be-haviour would require some manner of discontinuous change in the systemwhich would introduce dissipation at leading order in the 1N expansion . Sug-gestions were found that a potential source for such behaviour would be theworldvolume of the D-brane tending towards non-compactness in the IR inthe limit of high densities.This project allowed me to develop a thorough understanding of (non-)Fermiliquid physics and the machinery associated with treating fermions holography—in particular the subtle IR and large N issues associated with generating non-Fermiliquid behaviour in these models. From a numerical and computational perspectivesolving the integro-differential system and appropriately processing the solutionsrequired the development of a more sophisticated and efficient numerical frame-work then I had previously attempted. In addition scanning over large areas ofparameter space meant extensive use of Westgrid and the organizational overheadthis entails. These numerical and computational techniques were important in myfinal project, Chapter 6.From the perspective of the field this project successfully produced a novel,backreacted, holographic model of a Fermi liquid. This complements well the22existing work in the field. The works of pioneered in [47, 92] exhibit non-Fermibehaviour but suffer from large N artifacts. The Fermi liquid behaviour describedin [119] is free of large N artifacts but it is not clear how to begin to generalizethe setup to generate non-Fermi liquid behaviour. The efforts to create a fullybackreacted non-Fermi liquid in the bulk spacetime described in [5, 6] were veryambitious and while much progress was made the desired state was found to beinaccessible as a result of an apparent phase transition. An additional benefit ofthis project is that it shows that Minkowski D-brane embeddings can exist at finitedensities provided the matter content is fermionic. This addition to the probe D-brane toolkit will hopefully be useful in a variety of other contexts.1.8 Spatial Modulation and Conductivities in EffectiveHolographic TheoriesThis project was motivated by the desire to advance the understanding of holo-graphic models of inhomogeneity mediated, metal-insulator transitions 13. Thesefinite, but low, temperature transitions are controlled by parameters other then thetemperature (for example the periodicity of the sourced inhomogeneity) and repre-sent the finite temperature signatures of a quantum phase transition. An improvedunderstanding of the finite temperature behaviour would grant further insight intothe nature of the quantum phase transition, and indeed, the zero temperature groundstates themselves. The study of the PDEs governing such systems began in [72, 73].Here it was shown that the introduction of a sourced, ”lattice” inhomogeneity inone of the spatial directions via modulation of the chemical potential meant thatthe RN black hole could be generalized to include striped solutions. Once thesebackground solutions were known numerically the linearized perturbation equa-tions were derived and numerically solved in order to extract the Alternating Cur-rent (AC) electrical conductivity. The Direct Current (DC) conductivity was shownto be finite, unlike in the homogeneous case- see [54], and the low frequency ACbehaviour was seen to be well approximated by the Drude model of conductivity.In addition, an unusual scaling regime was seen to exist for a frequency range in the13As we explain the use of the word ”transition” here is heuristic. At finite temperature the behav-ior is a crossover with the true phase transition occurring only at zero temperature.23mid-IR which was reminiscent of the scaling regimes found in cuperate supercon-ductors. In [91] the exploration was extended to Einstein-Maxwell-Dilaton theory,again with inhomogeneity imposed by a modulated chemical potential, and againa mid-IR scaling regime for the AC conductivity was observed. Further analyticdevelopment in the Einstein-Maxwell theory model was undertaken in [39, 40].Their results allowed for the calculation of additional transport quantities- the ther-moelectric, electric and thermal conductivities as well as the electric conductivity,as a function of the near horizon behaviour of background solution. In additionvery high resolution and low temperature numerical calculations in these papersfound no evidence of any IR scaling regime.In our project we sought to extend the study of such transitions to the class ofEinstein-Maxwell-Dilaton theories considered in [20]. In these theories the gaugecoupling and potential function both have a free parameter which strongly influ-ences the IR behaviour of the solutions. In our study we chose to reduce the di-mension of our parameter space by choosing a particular linear relation betweenthese two parameters. This results in a single parameter which we label, υ . 14 Itwas shown in [20] that the range of IR profiles these functions exhibit in the ho-mogeneous case strongly suggests that both metallic and insulating phases shouldbe present once translational invariance is broken. Our goal was to generalize themethods used in [72, 73] and [39, 40] to numerically find these background solu-tions and to probe their transport properties.This undertaking required the derivation of the equations, and boundary andgauge conditions for the gravitational background and linearized perturbation equa-tions. In addition it was necessary to adopt the horizon paradigm of[39] to ourpurposes and to analyze the analytic properties of the resulting expressions. Nu-merical solution of these equations over large regions of parameter space and theefficient processing of the results required the implementation of efficient numeri-cal and data management techniques. This was achieved using the construction ofcode within, and interfaces between, Mathematica, Matlab and C++ libraries, andextensive use of the Westgrid computational facilities.14In their work Ling:2013nxa worked with a form of the potential and gauge coupling similar toours. Their functions, however, did not have any free parameters and they did not study the behaviorof this class of theories.24The results generated by this project include:• Using two distinct numerical techniques, involving solving both the back-ground and linearized perturbation equations in the bulk, we discover qual-itative changes in the DC electrical conductivity in the IR as a function ofthe period of the sourced inhomogeneity, k, and the parameter, υ . Thesechanges are strongly suggestive of the presence of metal insulator transitionin the zero temperature limit.• Using the solutions of the background equations and the horizon paradigmwe also show that this qualitative change in the physics can be seen in thethermoelectric conductivity and Lorenz factors but, curiously, not in the ther-mal conductivity.• We examine the presence of such a transition as a function of k and υ andobserve the existence of a non-trivial phase boundary with the possible exis-tence of incoherent metallic behaviour at the junction. In addition we illus-trate how the critical temperature for the onset of this qualitative change inphysics depends non-trivially on these two parameters and how the presenceof such a critical temperature is indicative of the redistribution of the spectralweight associated with the transition.• We probe the AC conductivity of the system using the solutions of the lin-earized perturbation equations and explicitly confirm this shift in the spectralweight in the regime where the transition occurs. In addition we illustratethat the mid-IR scaling observed in [73, 91] can exist as a function of param-eters but appears to be a fine-tuned and non-generic phenomena.From the point of view of the string theory community it is hoped that thisproject will aid in the continued effort to understand strongly coupled phase transi-tions from a holographic context. In particular, it is hoped that these finite temper-ature results will be useful in constructing the geometry describing to the metal-insulator quantum phase transition at zero temperature. The ability to constructsuch a solution would be a major step in confirming the value of holographic con-densed matter physics as an exploratory tool for identifying interesting new phasesof matter.25Chapter 2Holographic Higgs Phases2.1 Introduction and ConclusionsIn this chapter we examine the “spontaneous breaking of gauge invariance” fromthe perspective of the gauge-gravity duality. The duality has been utilized in con-texts in which we expect this phenomena to occur such as holographic supercon-ductivity (for a review see[69]) or color superconducting phases in QCD (for recentattempts to model such phases see [13, 21]).Of course, the expression “gauge symmetry” and its breaking is a misnomer,or more precisely relies on specific classical limit for its definition. In a specificweak coupling limit it makes sense to speak of gauge redundancies as approxi-mate global symmetries and use the machinery and language of global symmetrybreaking in this context. However, in an inherently non-perturbative context suchas holographic dualities one needs to stick to more precise and gauge-invariant def-initions. Such characterization of massive phases of gauge theories was given by’tHooft [67, 68], and we review this classification in Section 2.2.This classification of gauge theory phases is gauge invariant and non-perturbative,relying on the response of the gauge theory vacuum to massive external sources.This could be best used in the holographic context whenever we have an idea of thegauge theoretic microscopic definition of the system, and use the holographic con-text merely to perform calculations in the strongly coupled regime. This situationis demonstrated in Section 2.3, using one particularly simple such “top-down” con-26text, namely that of the Coulomb branch of the maximally supersymmetric SU(N)theory in four dimensions. We demonstrate that the phase structure of the theory ismanifested in certain geometrical features of the bulk theory which reproduce theexpected results.The purpose of this exercise is to extract a purely bulk criterion for the existenceof a Higgs phase interpretation of the theory, which we can then use in situationswhere the microscopic definition of the bulk theory is less well-understood. Indeed,we see that the expected behavior of the ’tHooft loop operator in the Higgs phaseimplies the existence of certain type of solitonic strings localized in the IR region ofthe bulk theory, representing a narrow magnetic flux tube in the boundary theory1.In the holographic context, this can be taken as the definition of such phases, sinceit implies much of the phenomenology we associate with the Higgs mechanism.Since our criterion depends only on the bulk geometry, it is ideal in the bottom-up approach to holographic duality, where the microscopic definition of the theoryis lacking. In Section 2.4 we demonstrate this criteria in the context of holographicsuperconductors, namely holographic theories with the Marolf-Ross prescription[100] (see also [136]) for obtaining boundary dynamical gauge fields (with finitegauge coupling). Such theories, in the broken phase, model genuine supercon-ductors rather than superfluids. We study the bulk and boundary properties of thesuperconducting vortices, and demonstrate their role in characterizing the phasestructure of the holographic theory.To this end, we construct new solitonic solutions in AdS4 black hole back-ground (in the probe limit), for various values of the boundary gauge coupling (theparameter α we introduce in Equation 2.4.6, by solving numerically the bulk equa-tion of motion – a set of coupled non-linear partial differential equation. Section 2.4is devoted to setting up the equations and boundary conditions, and describing theproperties of the solutions. Essential to our solutions is the use of dynamical bound-ary conditions for the gauge fields (introduced in [100]), which are necessary forobtaining finite energy solutions, corresponding to superconducting vortices2. We1The precise interpretation of this flux tube depends on the microscopic interpretation of thetheory, and in particular on the UV region of the geometry.2Superfluid vortices were constructed in [87]. Superconducting vortices, with infinite bound-ary gauge couplings, were constructed in [31]. We compare and contrast our solutions with thosesolutions below. See also [103] for a related construction.27describe in detail the bulk and boundary properties of our solution, and find a fewintriguing patterns in the dependence of their free energy on temperature and onthe boundary gauge coupling.We are hopeful that the criterion discussed here, and the role it plays in modelsof holographic superconductivity, will assist in formulating the problem of holo-graphic color superconductivity, and in constructing holographic models along thelines of [13]. We hope to return to this problem, one of the original motivations ofthe present note, in the near future.2.2 Characterization of Gauge Theory PhasesIn [67, 68] ’tHooft introduced a classification of phases of gauge theory based on itsresponse to electric and magnetic sources. For the characterization to be a precisedefinition of the associated phases, we restrict ourselves for now to theories withgauge group SU(N)/ZN , such as gauge theories based on unitary groups in whichall matter fields are in the adjoint representation. In such theories the centre of thegauge group ZN is a global symmetry which aids in providing order and disorderparameters to characterize the different phases.The response of the theory to electric sources is measured by the Wilson loopW (C) = Tr(ei∫C A) (2.2.1)where we take the trace in the fundamental representation. The curve C is taken torepresent the world line of two static external sources separated by distance L , andthe Wilson line then computes the static potential between these sources.The response to magnetic sources is similarly represented by a ’tHooft loopT (C), which plays a role of a disorder parameter in the theory. The ’tHooft loopoperator is defined in the path integral language as an integral over all gauge fieldconfigurations with a prescribed singularity along the curve C. The singularityrepresents the presence of an external magnetic sources. For the curve C whichrepresents the world line of two well-separated static sources, this operator probesthe theory in a way which is similar to the Wilson loop. Indeed, as is well-known,these two observables are exchanged under electric-magnetic duality (see for ex-ample [51] or section 10 of Witten’s lectures in [27]).28We can then distinguish the different phases3 of gauge theories by followingasymptotic behavior for large loops C:• Confinement W (C)∼ e−A(C) and T (C)∼ e−L(C)• Higgs Phase W (C)∼ e−L(C) and T (C)∼ e−A(C)We denote the area enclosed within the curve C by A(C) and the corresponding be-havior of the loop operator is called the area law. This encodes the linear potentialbetween the corresponding (electric or magnetic) sources. The linear potential hasan intuitive picture in terms of the existence of flux tubes connecting the sources(confining strings) which in turn exist because the corresponding (electric or mag-netic) flux lines emanating from the sources form narrow flux tubes and do notspread (the Meissner effect). Similarly, the length of the curve C is denoted byL(C), and the corresponding behavior for the loop operator is called the perimeterlaw. Such behavior encodes the fact that the fields generated by the correspond-ing source are short ranged (screened) and influence only the close vicinity of thesource location.In the Coulomb phase, or in a conformal field theory, the behavior of both theWilson and ’tHooft operators is dictated by conformal invariance. For the loopscorresponding to static sources separated by distance L, we have the behaviorW (C)∼ T (C)∼ e− aTL = e−TV (L) (2.2.2)where T is a large time cutoff, and a is a constant (which can depend on couplingconstants of the theory). Note that while formally this is classified as a perimeterlaw for both Wilson and ’tHooft loops, the behavior of the static potential V (L)in a massive (screened) phase is different V (L) ∼ e− LL0 where L0 is the screeninglength.The prescription of calculating the Wilson and ’tHooft loops in ADS/CFT issimple and well-known4. The expectation value of Wilson and ’tHooft loop re-spectively, in the fundamental representation and when working in the saddle point3This is not a complete classification of such phases. For example, there could be critical pointsand oblique confinement phases distinguished by the behavior of dyonic loop operators. We will notdiscuss such phases here.4We use in our holographic discussion the BPS loops, the so-called Wilson-Maldacena loop and29approximation, is of the form e−S. The action S is the minimal action of the world-sheet of fundamental string or D-strings respectively, in a configuration which endon the prescribed curve C on the boundary. As quantum operators the Wilson and’tHooft loops obey an interesting algebra which constrains the possible phases ofgauge theory, which was discussed in the context of ADS/CFT by Witten (section 5of [134]).The qualitative behavior of the Wilson loops in confining theories is also well-known. The electric flux tube connecting external sources is mapped into a stringworldsheet dipping into the bulk. In the confining phase the electric flux lines areconfined to narrow flux tubes. The dual statement is that the string worldsheetlocalizes in the bulk radial direction, oftentimes for clear geometrical reasons (e.g.the “end” of the IR geometry in some sense). In the next two sections we providean analogous statement for magnetic flux tubes in holographic theories in the Higgsphase5.2.3 Top Down ModelConsider k flat probe D3 branes in AdS5× S5 located at r = v (in Poincare coor-dinates), and smeared over the sphere S5. Here v is proportional to the VEV ofthe adjoint Higgs field giving rise to the Higgs mechanism in the N = 4 SYMtheory. This corresponds to the pattern of symmetry breaking SU(N)→ SU(N−k)×SU(k), in the large N limit, while k is kept finite6. In this example we have thepower of large N as an organizing principle, and we’ll see that it aids us in separat-ing the effects of symmetry breaking on the electric and magnetic loop operators.2.3.1 Electric flux linesWe are mainly interested in magnetic flux tubes, but we start with a brief discus-sion of the Wilson loop. In the broken phase, with the above breaking pattern,the static potential between electric sources in the fundamental representation isits magnetic dual [97, 113]. The asymptotic behavior for the loops we consider is unaffected by thepresence of the scalar fields. For suggestions on calculating the Wilson loop itself see [4].5For previous discussion of these flux tubes, see [75].6This is conventionally called to Coulomb phase, and indeed the leading order interaction be-tween electric sources will be Coulomb-like. Nevertheless we’ll use the term Higgs or broken phase.30schematically of the formV (L) =aL+bkNe−cvLLwhere a,b,c are constants. The leading order potential is still Coulomb-like, butsince k gauge bosons are now massive we have a 1N correction involving exchangeof those massive gauge bosons. This can be seen, for example, if we repeat thecalculations of [44, 45] in the broken phase.In the bulk this modification can be explained simply, as follows. The Wil-son loop calculation corresponds, in the saddle point approximation, to finding thearea of a fundamental string worldsheet whose boundary ends on the prescribedcurve C. The leading order term in the 1N expansion contributing to Equation 2.3.1corresponds to the calculation in pure ADS [97, 113]. The form of the leading1N correction in Equation 2.3.1 suggests a modification of the action of the samesaddle point.The required modification arises when considering the worldvolume theoryon the probe D3 branes. Consider the worldsheet of the fundamental string forwell separated electric sources, in the broken phase, a situation which is depictedin Figure 2.1. The leading order contribution for the Wilson line corresponds tothe worldsheet area, and 1N corrections come from interactions between the partof the worldvolume intersecting the probe branes (two lines on the probe branes,represented by two points in Figure 2.1 . Since the radial fluctuations of the probebranes are massive – those correspond to the longitudinal modes of the W-bosons– it is easy to see that exchange of the massive scalar fields corresponding to thesebrane fluctuations reproduce the form of the leading 1N correction in Figure 2.1.2.3.2 Magnetic flux linesIn contrast to the calculation of the Wilson loop outlined above, the ‘tHooft loopexpectation value changes character from perimeter to area law, already in the lead-ing order in the 1N expansion. This corresponds to the existence of a new type ofsaddle point, rather than a modification of the action of the existing worldsheet.The new saddle point is similar to that of the Wilson loop in confining theories.Indeed, in such case the geometry of the bulk provides an IR cutoff, such as a soft31BoundaryProbe BranesFigure 2.1: Wilson loop stretched between widely separated sources on theboundary. The leading order correction in the 1N expansion comes fromexchange of massive scalar representing the radial fluctuations of theprobe branes.or hard wall, or cap to the geometry. The area law is realized geometrically as theWilson line for widely separated electric sources receives contributions predom-inantly from the vicinity of the IR geometry. This is the holographic dual to thestatement that the flux lines connecting two electric sources do not spread out inthe confining vacuum.In our case the magnetic dual to that statement cannot be explained in termsof the bulk geometry alone. Indeed, the area law for the D1 brane has to arisefrom differences between the worldvolume theory of such brane and that of a fun-damental string (for example the different dilaton coupling [75]). In our simplemodel this is easy to identify: in the presence of the probe branes the worldvolumeof the D1 branes can take a detour through the probe D-branes which, for widely32BoundaryProbe BranesFigure 2.2: World volume of D1 brane stretched between widely separatedmagnetic sources on the boundary. The area law for the ’tHooft loopresults from the existence of a string-like object localized in the radialdirection. In this model such object can be represented as soliton on theworldvolume of the probe branes, drawn in a thick red line along theworldvolume of the probe brane.separated magnetic sources, will minimize the action. This is due to the fact thaton the worldvolume of the probe branes, the D1 brane can be transformed into asolitonic string of finite tension. Therefore asymptotically in such separation, theminimum action configuration would be the one depicted in Figure 2.2 in which theD-string worldvolume stretches mostly along the worldvolume of the probe branes.Note that this is qualitatively similar to the Wilson line in a confining theory, in thatthe radial location of the loop is stabilized at some fixed radial location for widelyseparated sources.The interpretation of the flux lines in this simple example depends in variousways on understanding the full gauge-gravity duality. In particular, we have used33large N scaling to distinguish electric from magnetic flux tubes, and correspond-ingly confinement from the Higgs mechanism. Furthermore, the microscopic in-terpretation of the theory helped identify the type of charges available in the gaugetheory, and which can be connected by those flux tubes. Nevertheless, we haveidentified a necessary condition for the existence of Higgs phase interpretation ofthe theory: the bulk spacetime should support a finite tension solitonic object whichis approximately localized in the radial direction. The existence of this object, dualto a narrow magnetic flux tube, is necessary for the ‘tHooft loop of the boundarytheory to obey an area law. In the next section we demonstrate the existence ofsuch solutions in a simple bottom-up model of holographic superconductivity.2.4 Application to Holographic SuperconductivityIn this section we discuss a specific 2+1 dimensional bottom-up model of holo-graphic superconductivity [58]. As argued above, an area law for the ’tHooft loopis guaranteed by a finite energy vortex solution of the bulk fields localized in theradial direction, representing magnetic flux tube in the boundary theory. When wedo not have a microscopic definition of the theory, we take the existence of suchsoliton as the definition of the Higgs phase in the bottom-up holographic context.We demonstrate below the existence of such finite energy solitons in the presentcontext.Crucial to the analysis is the prescription given in [100] (see also [136]) forobtaining dynamical gauge fields in the boundary theory, by requiring the bulkgauge fields to obey a specific type of boundary conditions in the UV, which we willrefer to as “dynamical“ boundary conditions. We show that with these boundaryconditions the required vortex solutions exist, and furthermore have finite energyper unit length. This indicates that the model, in the broken phase, describes agenuine superconductor.We then discuss the bulk and boundary properties of the solutions, includingthe dependence of their tension on the temperature and the boundary gauge cou-pling. Finally, the Higgs phase is characterized by electric screening, which wedemonstrate by examining the two point function of the boundary gauge field.342.4.1 The modelWe work in the context of the bottom-up model of [58]. The action is:S =12κ∫d4x√−g[R− 14F2µν −|(∂µ − iqAµ)ψ|2−V (ψ,ψ∗)]V (ψ,ψ∗) =6L2+m2ψψ∗ (2.4.1)where m2 < 0, and q is the charge of the scalar field. We work in the probe limit[58], defined as:q ∝ ε−1 Aµ ,φ ∝ ε ε → 0 (2.4.2)In this limit the matter energy momentum tensor scales as ε2 and drops out ofthe Einstein equation, and the metric is unaffected by the matter fields, while theMaxwell and scalar equations remain unchanged. The background solution fea-tures an ADS Schwarzschild black hole geometry which, for certain values of thethermodynamic variables (the chemical potential µ , or equivalently the tempera-ture T ) develops a profile for the scalar condensate and the temporal componentof the gauge field. This signals the onset of symmetry breaking below the criticaltemperature.We choose to work in cylindrical coordinates (t,ρ,θ ,w) with the conformalboundary located at w = 0. Our background metric is then:ds2 =L2w2(− f (w)dt2+ f (w)−1dw2+dρ2+ρ2dθ 2) (2.4.3)In addition we make use of the scaling symmetries of our action to scale the horizonto w+= 1 and we take the AdS radius of curvature to be L= 1. This fixes the metricfunction to be f (w) = 1w2 −w. By dimensional analysis we expect all physicalquantities to be proportional to the ratio of T/µ . In what follows we fix µ = 1 andexamine the behavior of the various quantities as a function of T .2.4.2 Ansatz and boundary conditionsWe now discuss matter excitations to the homogeneous background. Guided bythe known vortex solutions of the Abelian Higgs model in flat spacetime (reviewed35in appendix A), we propose the following ansatz for the solutions we seek:Aµ → (A0(w,ρ),0,Aθ (w,ρ),0)ψ → ψ(w,ρ)exp(isθ) (2.4.4)where s is the topological number associated with the vortex solution7.The equations of motion consist of the two Maxwell and one scalar equation:R2(qsw2 f− q2Aθw2 f)+(f ′f+2w)∂wAθ −∂ρAθw2ρ f+∂ 2ρAθw2 f+∂ 2wAθ = 0− q2A0R2w2 f+∂ρA0w2ρ f+∂ 2ρA0w2 f+∂ 2wA0 = 0R(q2A02w4 f 2− (s−qAθ )2w2ρ2 f+f ′w f− m2w4 f)+(f ′f+2w)∂wR+∂ρRw2ρ f+∂ 2ρRw2 f+∂ 2wR = 0(2.4.5)where we rescaled the scalar field as ψ(w,ρ)→wR(w,ρ), for reasons of numericalstability. It can be seen that in the probe limit described above there is a scalingsymmetry of the equations Equation 2.4.5, implying that if a solution is found for agiven value of q it is known for all q via an appropriate rescaling of the fields. Thisproperty is convenient for numerical purposes as it allows us to choose a scale forthe matter fields which is numerically tractable.We wish to solve our system of PDEs on the domain defined by w0 ≤ w ≤ w+and 0 ≤ ρ ≤ ∞, where w0 is a UV cutoff. For the problem to be well posed wemust choose self consistent boundary conditions which are also compatible withthe bulk equations of motion. We choose the following boundary conditions on thefour different segments of the boundary:• ρ→∞: In flat space it is known that the vortex fields decay exponentially to-wards asymptotic values for the gauge and scalar fields as ρ goes to infinity.Anticipating similar behavior, in our numerical implementation we imposea Neumann boundary conditions at some finite and large value, ρcut , since in7Of course, this number is not conserved in the full geometry, and indeed as we will see it “un-winds“ as function of the bulk radial coordinate w.36that region the solution should tend to the homogeneous ground state8.• ρ → 0: To determine the boundary conditions at the vortex core we requirethat all components of the bulk magnetic field be finite. The radial andtransverse components of the magnetic field are given by Bw = ∂ρAθ/ρ , andBρ = ∂wAθ/ρ , respectively. Finiteness of the radial component implies that∂ρAθ → 0 as ρ → 0. Regularity of the transverse component then restrictsthe Aθ component to obey (in this limit) ∂wAθ → 0. Therefore we concludethat Aθ (w,ρ = 0) must be a constant. If we were to impose a Dirichlet con-ditions at the conformal boundary this would fix this constant to be zero. Inour case we have a residual gauge freedom9, consistent with the boundaryconditions, which we use to set Aθ (w,ρ = 0) = 0 at the core of the soliton.• w→ w0 : On the conformal boundary we impose Dirichlet conditions on thescalar and A0 fields — the scalar field must be normalizable and A0 mustasymptote to the chemical potential µ . Crucially, on the Aθ field we imposethe following boundary condition∂Aθ∂w= α ρ∂∂ρ(1ρ∂Aθ∂ρ) (2.4.6)at the conformal boundary. This corresponds to having a theory in which theboundary value of the bulk gauge field corresponds to a gauge field [100] inthe boundary theory10. The parameter α determines the gauge coupling e2of the boundary gauge field, e2 = g2bulk/α . Indeed, having a consistent varia-tional principle requires the addition of the boundary action to Equation 2.4.1:Sbdy =1e2∫d3x√−hF2 (2.4.7)where the integration is over the boundary whose induced metric is denoted8ρcut is chosen such that our solutions vary by less than 0.01% if it is increased.9The dynamical boundary conditions at the conformal boundary allow for gauge transformationswhose parameter is independent of the radial coordinate w.10We choose to make dynamical only the component Aθ of the gauge field, for the sake of sim-plicity, we do not expect the features of the solution to change much if A0 is made dynamical aswell, since it already has nearly vanishing radial derivative near the conformal boundary, in all thesolutions we are interested in.37by h. F is the field strength for the boundary gauge field. We refer to theseboundary conditions as the “dynamical” boundary conditions in what fol-lows.• w→ w+: Regularity conditions at the horizon are necessary since the equa-tions degenerate there. Choosing the solutions which are regular at the hori-zon means that the coefficients of the divergent terms in a power series ex-pansion of the equations near the horizon have to vanish. This prescriptionyields the following constraints in our case:A0 = 0R(−q2Aθ 2ρ2+2qsAθρ2−m2− s2ρ2−3)−3∂wR+ ∂ρRρ +∂2ρR = 0 (2.4.8)R2q (s−qAθ )−3∂wR−∂ρAθρ+∂ 2ρAθ = 0We numerically solve the equations with these boundary conditions using suc-cessive over-relaxation (SOR) in the domain w0 ≤w≤w+ and 0≤ ρ ≤ ρcut , wherethe truncation radius ρcut is large but finite. In this approach the equations are dis-cretized on a lattice covering the domain of integration. We use a second orderfinite differencing approximation in which the derivatives are replaced with theirfinite differencing counterparts. An initial guess for the value of the scalar andgauge fields is then assigned to each grid point. Dirichlet boundary conditionsare implemented by insisting that the initial values assigned to the fields at theboundary grid points are maintained throughout the relaxation procedure, whereasNeumann or Robin boundary conditions must be imposed after each iteration. Thisis done by using the discrete form of the derivative operators to update the bound-ary grid points based on the values calculated for the interior points. The SORalgorithm then provides an iterative method of finding numerical solutions to thisfinite difference system to within a prescribed tolerance. Once the solutions areavailable other quantities of interest such as the energy density are calculated viainsertion of these solutions into the suitably discretized action. Further details ofour implementation are found in Appendix B.382.4.3 Free energyBefore presenting the numerical solution and discussing its properties, we explainthe reason we expect the energy (per unit length) to be finite in our case. Thediscussion parallels that of [87].The Lagrangian density of the bulk fields is:L = q2ρA02R22 f −R2((s−qAθ )22ρ +12 w2ρ f + m2ρ2)− 12 w4ρ f (∂wR)2−w3ρ f R∂wR−12 w2ρ(∂ρR)2+ρ(∂ρA0)22w2 f +12ρ(∂wA0)2− w2 f (∂wAθ )22ρ −(∂ρAθ )22ρ (2.4.9)Since the resulting action diverges near the boundary, we regularize it by sub-tracting the action of the translationally invariant hairy black hole solutions fromthe on-shell vortex action. Such subtraction automatically removes the divergenceswhich occur due to integration in the w direction. Therefore divergences, if theyexist, can occur only as a result of ρ integration. In the region of large ρ the scalarand A0 fields asymptote to their values in the translationally invariant ground state.Therefore the only terms in the (asymptotic) Lagrangian density to survive the sub-traction procedure are:−2pi∫dt∫ 10dw∫ ρcut0dρ[R2(s−qAθ )22ρ+w2 f (∂wAθ )22ρ](2.4.10)where all fields are understood to be functions of w only. Here we have introducedthe cutoff ρcut in order to regulate potential divergences in the ρ integration.If we now use the Aθ equation of motion to make the substitution:−12R2(s−qAθ ) = w(w f′+2 f )∂wAθ2q+w2 f∂ 2wAθ2q(2.4.11)and integrate by parts, using the fact that f vanishes on the horizon, we obtain thelogarithmically divergent term:pi log(ρcutγ)∫dt(w2 f Aθ∂wAθ |w=0 +(sq)∫ 10dw∂w(w2 f∂wAθ ))= pi log(ρcutγ)∫dt(w2 f Aθ∂wAθ − sqw2 f∂wAθ)|w=0 (2.4.12)39In integrating by parts we have introduced the length scale γ which is a measure ofthe size of the vortex core.This reasoning led the authors of [87] to conclude that their vortex solutionis logarithmically divergent, as expected from vortices in a superfluid. We seethat if we instead consider dynamical boundary conditions for the Aθ field, then∂wAθ |w=0 = 0 outside the core of the soliton. Then, provided an appropriate vortexsolution exists, the coefficient of the logarithmic divergence will vanish. We seebelow that indeed such vortex solutions (whose profile significantly differs fromthe superfluid vortices found in [87]) do exist and we calculate their finite energy(per unit length). This demonstrates that our model describes a genuine supercon-ductor11.2.4.4 Bulk properties of the solutionsWe are now ready to present our numerical solutions for the bulk fields and discusstheir properties for different values of the parameter α . We leave discussion of ournumerical solution to appendix B.The system has a critical temperature Tc, below which it is in the condensedphase (i.e. the scalar field develops a normalizable background). Below that criticaltemperature vortex solutions start appearing, in Figure 2.3 we show the profileof the fields for a typical vortex solution for α = 0. The form of the solutionsmay be understood as follows: far from the vortex core the fields tend to theirhomogeneous profiles and the PDEs reduce to ODEs. When solving these ODEsnumerically one finds that the solution for the Aθ field is a constant, given bysq . Together with our previous discussion of the ρ → 0 boundary conditions, thismeans that Aθ asymptotes to a constant, independent of the radial coordinate, bothas ρ→ 0 and as ρ→∞. Since we are also demanding vanishing radial derivative atthe conformal boundary, a reasonable guess is that the global Aθ solution dependsonly on ρ , i.e. Aθ (w,ρ)→ Aθ (ρ). This is indeed what we find numerically. Asseen in Equation 2.4.12 above, the asymptotic form of Aθ is directly responsiblefor the finiteness of the vortex energy.We have also obtained the solution with α 6= 0, in other words with dynamical11This was shown for α = 0 in [31].4000.20.40.60.8102040608000.020.040.060.080.10.12wR(w,ρ)ρ 00.20.40.60.8102040608000.020.040.060.080.1wAθ(w,ρ)ρ00.20.40.60.8 102040608000.0050.010.0150.020.025wA0(w,ρ)−µ (1−w)ρFigure 2.3: The matter field profiles at a temperature of ' 0.89Tc In orderto aid in visualization the background translationally invariant solutionhas been subtracted from the A0 gauge field. Note the asymptotic ap-proach of the scalar and A0 fields to their translationally invariant pro-files and the fact that Aθ field is independent of the radial coordinate w,and asymptotes to sq as ρ → ∞.boundary gauge fields. In Figure 2.4 we demonstrate the effect of the boundaryaction by displaying the differences in bulk fields (relative to the α = 0 case) forthe specific case of α = 3. It can be seen that the profile of the fields is no longerhomogeneous in the w direction near the core of the vortex. The greatest inho-mogeneity is seen in the Aθ and A0 fields while the changes in the R field, whilesubstantial in magnitude, are largely homogeneous in w.Once we obtain the numerical solutions for the matter fields, their on-shellaction can be evaluated. In Figure 2.5 we illustrate the profile of the free energy41Figure 2.4: The additional contribution to the bulk fields resulting from theaddition of a boundary action with α = 3. The plots are normalizedwith respect to the α = 0 profiles. It can be seen that, as expected, thegreatest variation is seen in the Aθ field.density of both the translationally invariant and vortex solutions for α = 0, and theirdifference. We note that the bulk free energy density of the vortex solution, in thevicinity of the core of the soliton, dips below that of the homogeneous ground statenear the conformal boundary. Nevertheless, as we will see below the boundary freeenergy density of the vortex (relative to the background) is everywhere positive.We next turn to solutions with α 6= 0. In Figure 2.6 we plot the profile ofthe bulk energy density (with the homogeneous background subtracted) for theα = 3 solution, and the difference between that solution and the α = 0 solution.We see that the change in the free energy density can be significant and is heavilylocalized near the conformal boundary and the core of the vortex. We also note42Figure 2.5: Free energy density profiles for the translationally invariant back-ground and vortex solution, and their difference for α = 0 and a tem-perature of ' 0.91Tc. Note for ease of visualization we have includedthe w factors coming from the measure.that, as expected, increasing α has the effect of shifting more of the contributionof the action to the vicinity of the conformal boundary at the expense of the bulk.2.4.5 Boundary properties of the solutionsThe boundary free energy density can be found by the standard procedure of inte-grating radially the Euclidean on-shell action, and including both the countertermaction and the boundary Maxwell term (whose coefficient is α). We now discussthe boundary free energy and its dependence on various parameters.In Figure 2.7 we display the boundary free energy density for several values43Figure 2.6: Bulk free energy for the vortex solution with α = 0, and thedifference between this and the α = 3 solution at a temperature of' 0.91Tc. We note the increased energy density near the conformalboundary relative to the α = 0 case.of the temperature (at α = 0.001). At low temperatures (relative to the criticaltemperature) one sees that the vortex energy profiles are sharply peaked near ρ = 0and that, as one approaches the critical temperature, they flatten and broaden asthe vortices begin to disperse. The vortex solutions merge with the homogeneousbackground at the critical temperature Tc.Once we make the boundary gauge fields dynamical (i.e. turn on α), the so-lutions significantly change and the free energy receives additional contributionsfrom the boundary Maxwell action. In Figure 2.8 we plot the total boundary freeenergy for various values of the coupling α . We see that the energy density is thatof a finite size lump, as expected, and that turning on α can be quite significant atthe core of the vortex, for the range of couplings displayed.2.4.6 Dependence on parametersIn order to display the dependence of the total boundary free energy on temper-ature, in Figure 2.9 we show the decrease in the free energy as we approachedthe critical temperature Tc (found from examining the onset of the translationallyinvariant condensate). Fitting the curve to a function of the formF = α(1−T/T c)β (2.4.13)440 0.5 1 1.5 2 2.5 300.0050.010.0150.020.0250.030.0350.04ρFree energyBoundary free energy density  T1=0.98329T2=0.95124T3=0.90547T4=0.84966T5=0.78781Figure 2.7: Boundary free energy density of the soliton for several valuesof the temperature relative to the critical temperature, Tc. Notice thechanges in the vortex profile as a function of temperature — at lowtemperatures it is peaked near the vortex core while as the temperatureincreases it tends to become wider and more diffuse, tending to the ho-mogeneous background at Tc.yields approximately α = 0.0529,β = 1.0637. In other words, up to numerical in-accuracies, the free energy of the soliton coincides with that of the translationallyinvariant (uncondensed) background at the critical point, and depends on tempera-ture approximately linearly in the low temperature phase.It is also interesting to examine the string tension (which corresponds to the in-ternal energy), which quantifies the strength of magnetic confinement, as functionof temperature. We exhibit that dependence in Figure 2.9 as well, we see that thequalitative behavior is similar to that of the free energy. We note that the fact thatthe free energy goes to zero in an (approximately) linear fashion as one approachesthe critical temperature ensures that the vortex solutions appear initially with somefinite internal energy.In Figure 2.10 we display the dependence of the total boundary free energy onthe parameter α . The dependence we find is intriguing: as we increase α (corre-450 0.5 1 1.5 2 2.5 300.0020.0040.0060.0080.010.0120.0140.016ρFree energyBoundary free energy density  α1=0α2=0.75α3=3Figure 2.8: Profile of the boundary free energy density of the vortex, forvarious values of the boundary gauge coupling α , at a temperature of' 0.91Tc. We note that the energy density as a function of α quicklybegins to saturate.sponding to decreasing the boundary gauge coupling) the free energy rises rapidlyand eventually saturates, resulting in finite free energy difference between α = 0and α → ∞. Fitting to a function of the form12F = Aexp(−B/α)+C (2.4.14)yields approximately A = 0.009,B = 0.257,C = 0.004. The exact interpretation ofthis result is unclear. We note that the large α limit corresponds to taking e2 to zero(as the bulk coupling must be kept small in order for classical gravity to be valid.)Naively this would lead one to believe that the boundary term in the bulk gravityaction, and the corresponding term in the field theory partition function, becomefree Maxwell theories. However as the boundary action serves to implement theboundary conditions for the bulk equations of motion and the gauge field in the field12This form is consistent with the existence of a perturbative expansion in the boundary gaugecoupling.46theory is an emergent component of a strongly coupled system this interpretationis probably incorrect. It would be interesting to investigate this issue further.2.4.7 Electric screeningFinally, for the sake of completeness we comment on the behavior of the vac-uum in the presence of electric sources. Instead of probing the response to thosesources by calculating the Wilson line, it is simpler in our case to concentrate onthe Green’s function of the boundary gauge field. While in the case of Dirichletboundary condition the Green’s function encodes the optical conductivity, in thecase of dynamical boundary conditions this encodes the electric response of thesystem. In order to demonstrate the expected behavior of electric screening, wehave to show that the static (zero frequency) long distance limit of the Green’sfunction is gapped. We demonstrate the gap in 2.11 by displaying the low momen-tum limit of the zero frequency Green’s function. This clearly stays bounded as wetake the zero momentum (long distance) limit.2.5 ConclusionIn conclusion, in this chapter we constructed vortex solutions in the context of theholographic models of [58], for various values of the bulk and boundary parame-ters. These vortices signify the onset of local symmetry breaking. The impositionof the dynamical boundary conditions corresponds, via the prescription of [100], toa dual field theory with dynamical gauge field, with varying values of the bound-ary gauge coupling. This is evidenced, for example, by the fact that any boundarygauge transformation which is only a function of the boundary coordinates re-spects the dynamical boundary conditions on the bulk gauge field. We find thatin the spontaneously broken phase, the symmetry breaking is manifested by theexistence of bulk vortex solutions with the expected properties of superconductingvortices: there is no operator corresponding to a superfluid current on the boundary,and the vortex boundary energy is finite. In contrast, as found at [87], the impo-sition of Dirichlet boundary conditions leads to a theory which exhibits a globalsymmetry breaking and vortices with diverging energy, as expected in a superfluid.We expect that the criteria developed here for characterizing local and global47symmetry breaking will have applications in other bottom-up holographic models.In particular it would be interesting to explore the applicability of these techniquesto models of finite density QCD and color superconductivity.480 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0800.511.522.533.54x 10−3 Free energy near critical point1−T/TcFree energy0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.0420.0430.0440.0450.0460.0470.0480.049Internal energy of vortex solutions near critical point1−T/TcFree energyFigure 2.9: The total boundary free energy and the string tension (internal en-ergy) as a functions of temperature below the critical temperature. Thenearly linear behavior is in agreement with the Landau-Ginzburg modelof superconductivity near the critical point. However as these solutionsare normalized with respect to the translationally invariant condensatethe fact that the linear behavior continues to exist for the vortex solu-tions is noteworthy.490 0.5 1 1.5 2 2.5 345678910111213x 10−3 Free energy as a function of couplingαFree energyFigure 2.10: Boundary free energy density as function of the boundary gaugecoupling α . As expected the free energy remains bounded in the limitthat α is taken to be small or large.2 4 6 8 10k0.20.40.60.8Greens' Function2-point Greens functionFigure 2.11: Zero frequency two point function of the boundary gauge fieldin the limit of small momentum.50Chapter 3Holographic Stripes3.1 Introduction and SummaryIn this chapter we apply gauge-gravity duality to study the spontaneous breakingof translation invariance and the formation of striped order. Stripes are known toform in a variety of strongly coupled systems, from large N QCD [30, 121] tosystems of strongly correlated electrons (for a review see [129]). The formationof stripes and the associated reduced dimensionality are speculated to be related tothe mechanism of superconductivity in the Cuprates [19]. It is therefore useful tostudy striped phases in the holographic context.Besides its interest in the boundary theory, this study has an intrinsic interestin the bulk gravitational context1. We describe striking bulk and boundary prop-erties of our bulk solutions, including frame dragging effects, the magnetic field,the curvature and the geometry. Some of the features can be understood as theemergence of a near horizon region which acts as a bulk topological insulator. Themagnetoelectric effect is then responsible for the patterns we observe for the bulkmagnetic field and vorticity.Our study is facilitated by a numerical solution of the set of coupled non-linearEinstein and matter equations in the bulk, which exhibit a normalizable inhomoge-1Our model describes a black hole whose instability to the formation of inhomogeneous structuresresembles the black string instability [53] which is known to be of the second order for high enoughdimensions [123].51neous mode. Previous studies of inhomogeneous solutions in asymptotically ADSspacetimes concentrated on non-normalizable modes [49] (i.e. explicit rather thanspontaneous breaking of translation invariance) or the study of co-homogeneityone solutions [35, 36, 78, 104], where one of the translational Killing vectors is re-placed by a helical Killing vector. More recently, such spontaneous breaking wasexhibited in a probe model, which was shown to have a magnetic field induced lat-tice ground state [18]. In contrast to the above, our solutions are co-homogeneitytwo, they backreact on the geometry, and exhibit spontaneous breaking of transla-tion invariance below a critical temperature.These features are analyzed as a function of temperature. In particular, we findthat the horizon of the black hole develops a “neck” and a “bulge” in the transversedirection which shrink with temperature, such that the ratio of their sizes contractsas fast as∼ Tσ , with an order σ ∼ 0.1 exponent. Simultaneously, the proper lengthof the horizon in the transverse direction grows at a rate ∼ 1/T 0.1. However, thecurvature remains finite, and its maximal value, occurring at the bulge, tends to aconstant in the limit T → 0.The bulk black hole solutions give rise to the holographic stripes on the bound-ary, characterized by non vanishing momentum and electric current and modula-tions in charge and mass density. Starting small near Tc, the amplitudes of the mod-ulations grow steadily at lower temperatures, approaching finite values at T → 0.Finally, we study the thermodynamics of the system by constructing phase dia-grams in various ensembles. For small values of the axion coupling, where the ther-modynamic potentials in both phases are nearly degenerate, our numerical methodis not accurate enough to sharply distinguish between weak first order and secondorder transitions 2. However, for sufficiently large values of the axion coupling wediscover a clear second order phase transition in the canonical (fixed charge), thegrand canonical (fixed chemical potential) and the micro-canonical ensembles. Wedescribe both the finite system (of fixed length) and the infinite system, where wefind that the dominant stripe width changes as function of temperature.2When the revised version of this Letter was nearly ready to be submitted the preprints [32] ap-peared, investigating this model using different methods. These preprints also indicate the existenceof a second order phase transition for sufficiently large axion coupling.523.2 The Holographic SetupThe Lagrangian describing our coupled system is [34]L =12R− 12∂ µψ∂µψ− 14FµνFµν − V (ψ)−Lint ,V (ψ) = −6+ 12m2ψ2,Lint =1√−gc116√3ψ εµνρσFµνFρσ , (3.2.1)where R is the Ricci scalar, Fµν is the Faraday tensor, Lint describes the axioncoupling and g is the determinant of the metric. We use units in which the ADSradius l2 = 1/2, Newton’s constant 8piGN = 1, and c= h¯= 1, and choose m2 =−4and several values of c1.Perturbative instabilities towards the formation of charge and current densitywaves were identified in [34] for a range of wave numbers and temperatures3. Wenote the appearance of axion electrodynamics in the bulk theory. It is curiousthat inhomogeneous instabilities (see also [104]) seem to involve the topology ofthe bulk fields in an essential way, though the analysis performed to discover theinstability is local in nature. We leave this mystery for future work.In this Letter we investigate the end-point of the instability. Part of the bound-ary data is the spatial periodicity, and we focus mostly on the wave number withthe largest critical temperature Tc [34]4. This state is a co-homogeneity two so-lution, thus we construct the family of stationary solutions that emerge from thecritical point, assuming all the fields to be functions of the radial coordinate r andone spatial coordinate x.Our ansatz includes the scalar field ψ(r,x), the gauge field components At(r,x)3 An interesting application of the instability in this model has appeared very recently [42].4Alternatively one can work in the conjugate ensemble where the tension in the spatial directionis fixed. We found that the order of the phase transition does not change in this case, details willappear elsewhere [118].53and Ay(r,x) and the metricds2 = −2r2 f (r)e2A(r,x)dt2+2r2e2C(r,x)(dy−W (r,x)dt)2+ e2B(r,x)(dr22r2 f (r)+2r2dx2), (3.2.2)where for the sake of convenience we included in the definition of the metric func-tions the factor f (r) characterizing the metric of the RN solution, with horizon atr = r0:f (r) = 1−(1+µ24r20)(r0r)3+µ24r20(r0r)4.The inhomogeneous solutions reduce to the RN solution above the critical temper-ature.The conformal in r,x plane ansatz (3.2.2) is convenient in constructing co-homogeneity two solutions. With this ansatz, the Einstein and matter equationsreduce to seven coupled elliptic equations and two constraint equations. More-over, the constraint system can be solved elegantly using its similarity to a Cauchy-Riemann problem [131].The boundary conditions we impose correspond to regularity conditions at thehorizon and asymptotically ADS conditions at the conformal boundary. With theseboundary conditions, the set of solutions we find depend on three parameters: thetemperature T , the chemical potential µ and the periodicity in the x direction L.Using the conformal symmetry inherent in asymptotically ADS spaces, the modulispace of solution depends only on the two dimensionless combinations of theseparameters. To focus on the dominant critical mode that becomes unstable at thelargest temperature Tc we choose L = 2pi/kc.On the spatial boundaries it is useful to impose “staggered” periodicity con-ditions. Using two reflection symmetries which are preserved by the form of theunstable perturbation, one can reduce the numerical domain to a quarter period andimpose5 ∂xψ(x = 0) = 0, ψ(x = L/4) = 0, h(x = 0) = 0, ∂xh(x = L/4) = 0 and∂xg(x = 0) = 0, ∂xg(x = L/4) = 0, where h represents the fields Ay and W , and g5Our boundary conditions do not exclude the homogeneous solution, but since that solution isunstable we find that in practice our numerical procedure converges to the inhomogeneous solutionunless we are very close to the critical point.54Figure 3.1: Metric functions for θ ' 0.11 and c1 = 4.5. Note that the metricfunctions A,B and C have half the period of W . The variation is maxi-mal near the horizon, located at ρ = 0, and it decays as the conformalboundary is approached, when ρ → ∞. The matter fields (not shown)behave in a qualitatively similar manner.refers collectively to A,B,C and At .The elliptic equations derived from (3.2.1) are discretized using finite differ-ence methods and are solved numerically by a straightforward relaxation with thespecified boundary conditions. In this method the equations are iterated startingwith an initial guess for all fields, until successive changes in the functions dropbelow the desired tolerance. We verify that the remaining two constraints are sat-isfied by those solutions. Full details of our procedure are given in the upcoming[118].553.3 The SolutionsA convenient way to parametrize our inhomogeneous solutions is by the dimen-sionless temperature θ = T/Tc, relative to the critical temperature Tc. Our methodallows us to find solutions in the range 0.003. θ . 0.9 for c1 = 4.5 and the range0.00016. θ . 0.96 for c1 = 8, for fixed µ .Bulk Geometry. For subcritical temperatures, as we descend into the inhomoge-neous regime, the metric and the matter fields start developing increasing variationin x. Fig. 3.1 displays the metric functions for θ ' 0.11, over a full period in thex direction, in the case c1 = 4.5. The matter fields have qualitatively similar be-haviour. The variation of all fields is maximal near the horizon of the black holeat ρ ≡√r2− r20 = 0, and it gradually decreases toward the conformal boundary,ρ → ∞.Many of the special features of the solutions we find are related to the presenceof axion electrodynamics, the effective description of the electromagnetic responseof a topological insulator, in the gravity action. In the broken phase we have anaxion gradient in the near horizon geometry, which therefore realizes a topologicalinsulator interface6. The presence and the pattern of a near horizon magnetic field,summarized in the field Ay, can be related to the magnetoelectric effect in suchinterfaces.In curved space the magnetic field is accompanied by vorticity, which is man-ifested by the function W . This causes frame dragging effects in the y direction.Test particles will be pushed along y with speeds W (r,x), in particular the directionof the flow reverses every half the period along x. The drag vanishes at the hori-zon and at the location of the nodes of W where x = Ln/2, for integer n (see Fig.3.1). In general, the dragging effect remains bounded, the vector ∂t is everywheretimelike, and no ergoregion forms.The Ricci scalar of the RN solution is RRN =−24, constant in r and independentof the parameters of the black hole. This is no longer true for the inhomogeneousphases, where the Ricci scalar becomes position dependent. The right panel of6It would be interesting to discuss localized matter excitations on the interface, especiallyfermions, along the lines of [114].56Figure 3.2: Left panel: The variation along x of the size of the horizon inthe y direction includes alternating “necks” and “bulges”. Right panel:Ricci scalar relative to that of RN black hole, R/RRN −1 for θ ' 0.003over half the period. The scalar curvature is maximal along the horizonat the bulge x = nL/2 for integer n. The axion coupling here is c1 = 4.5and similar results appear for other c1’s.Fig. 3.2 illustrates the spatial variation of the Ricci scalar, relative to the RRN forθ ' 0.003. The plot corresponds to c1 = 4.5, however we observe qualitativelysimilar results for other values of the coupling.The maximal curvature is always along the horizon at x= nL/2 for integer n. Itgrows when the temperature decreases and approaches the finite value of R'−94in the small temperature limit.The left panel in Fig. 3.2 shows the variation of transverse extent of the horizonin the y direction, ry(x) ≡√2r0 exp[C(r0,x)], along x for θ ' 0.003. Typicallythere is a “bulge” occurring at x = nL/2 and a “neck” at x = (2n+ 1)L/4, forinteger n. Note that Ricci scalar curvature is maximal at the bulge and not at the57neck as would happen, for instance, in the spherically symmetric black string case.The size of both the neck and the bulge monotonically decrease with temperature,however, the neck is shrinking faster. We find that the ratio scales as a powerlaw rnecky /rbulgey ∼ θσ near the lower end of the range of θ ’s that we investigated.The exponent σ depends on the coupling, ranging from about 0.5 for c1 = 4.5 toapproximately 0.1 for c1 = 8.Another aspect of the geometry is the proper size of the stripe in the x direc-tion at fixed r, lx(r)≡∫ L0 exp[B(r,x)]dx. The proper length tends to the coordinatelength as 1/r3 asymptotically as r→ ∞, but it exceeds that as the horizon is ap-proached. Namely, the inhomogeneous phase “pushes space” around it along x,resembling the “Archimedes effect”. The proper length of the horizon is maximaland it grows as the temperature decreases. We find that at small θ the proper lengthof the horizon diverges approximately as ∼ θ−0.1.Boundary Observables. Near the conformal boundary the fields decay to theirADS values, and the subleading terms in their variation are used to define theasymptotic charge densities of our solutions. The subleading fall-offs of the metricfunctions in our ansatz determine the boundary stress-energy tensor, whereas thefall-offs of the gauge field determine the charge and current densities of the bound-ary theory. Finally, the subleading term of the scalar field near infinity determinesthe scalar condensate.For our inhomogeneous solutions we find that all charge and current densi-ties are spatially modulated, except for 〈Txx〉, which is constant, consistent withthe conservation of boundary energy-momentum. We define the total charges of asingle stripe by integrating the charge densities over the full period L. These inte-grated quantities are charge densities per unit length in the translationally invariantdirection y.3.4 ThermodynamicsWe demonstrated that below the critical temperature Tc there exists a new branchof solutions which are spatially inhomogeneous. The question of which solutiondominates the thermodynamics depends on the ensemble used. We start our dis-58-0.3-0.2-0.10.0HF-F RNLN20.0 0.2 0.4 0.6 0.8 1.0TTc-0.04-0.020.00HW-WRNLΜ2Figure 3.3: Difference in the thermodynamic potentials between the inhomo-geneous phase and the RN solution for c1 = 8, plotted against the tem-perature. In both ensembles there is a second order phase transition,with the inhomogeneous solution dominating below the critical temper-ature.cussion by fixing the boundary periodicity, corresponding to working in a finitesystem of length L7 . We discuss the system with infinite length in the inhomoge-neous x-direction below.The canonical ensemble corresponds to fixing the temperature and the totalcharge. This describes the physical situation in which the system is immersed ina heat bath consisting of uncharged particles. In the upper panel of Fig. 3.3 weplot the difference of the normalized total free energy, F = M−T S, between thetwo classical solutions as function of the temperature T , for c1 = 8. In our ensem-ble the total charge N is fixed, and we use the scaling symmetry of the boundarytheory to set N = 1, or in other words measure all quantities in terms of N. Asa result the free energy is a function of one parameter, the temperature T . The7We mostly discuss the case L = 2pikc , where kc is the wavelength of the dominant instability, thatwith the highest critical temperature. Results for other values of L will appear elsewhere [118], andare qualitatively similar.59Extremal RN0.72 0.74 0.76 0.78 0.80 0.82 0.84MN20.00.51.01.52.02.5SNFigure 3.4: The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). Below the criticaltemperature, the striped solution has higher entropy than the RN. TheRN branch terminates at the extremal RN black hole, while the stripedsolution persists to smaller energies.figure displays a second order phase transition, where the inhomogeneous solutiondominates the thermodynamics below the critical temperature Tc, the temperatureat which inhomogeneities first develop.If we fix the chemical potential instead of the charge, we discuss a situationwhere the system is immersed in a plasma made of charged particles. To studythe thermodynamics we use the grand canonical free energy Ω = M−T S− µN,displayed in the lower panel of Fig. 3.3. In this ensemble it is convenient to mea-sure all quantities in units of the fixed chemical potential µ . Then, again, the freeenergy is a function of only the temperature T . In the fixed chemical potential en-semble we find a similar second order transition, where the inhomogeneous chargedistribution starts dominating the thermodynamics at the temperature where theinhomogeneous instability develops.The physical situation relevant to the study of the real time dynamics of the60instability corresponds to fixing the mass and the charge. This is the microcanon-ical ensemble, describing an isolated system in which all conserved quantities arefixed. In this ensemble it is convenient to measure all quantities in terms of the(fixed) charge, and the remaining control parameter is then the mass M. We findthat in this ensemble as well, the striped solutions dominate the thermodynamics(have higher entropy) for all temperature below the critical temperature Tc, at leastwhen the axion coupling c1 is sufficiently large. This is shown in Fig. 3.4.Finally, we can also study the infinite system in the inhomogeneous x-direction,which we choose to look at in the canonical ensemble. In this case we are in aposition to compare the free energy density of different stripes, of different lengthsin the x-direction. This comparison is shown in Fig. 3.5, where we see that thequalitative picture is the same as in the finite system – a second order transition withstriped solutions dominating at every temperature below the critical temperature.Just below the critical temperature, the dominant stripe is that corresponding tothe critical wavelength kc. However, for lower temperature different stripes willdominate, in fact we see in Fig. 3.5 that the dominant stripe width tends to increasewith decreasing temperature.611.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10LΜTΜFigure 3.5: A contour plot of the free energy density, relative to the homoge-nous solution. The red line shows the variation of the dominant stripewidth as function of the temperature for c1 = 8.62Chapter 4Striped Order in AdS/CFT4.1 Introduction and SummaryAs described in Chapter 1 the gauge-gravity duality has shone new light on manycondensed matter systems - see [54, 65, 102, 120] for reviews. Early models inthis area, such as the holographic superconductor [59], focused on homogeneousphases of field theories. In this case, the fields on the gravity side depend onlyon the radial coordinate in the bulk and the problem reduces to the solution ofODEs. However, many interesting phenomena occur in less symmetric situations.Generically, the problem of finding the gravity dual to an inhomogeneous bound-ary system will necessitate solving relatively more difficult PDEs, almost alwaysresulting in the need for numerical methods. While these become technically hardproblems, there exist established numerical approaches. Due to the success of theholographic method in studying homogeneous situations, it is worthwhile to pushthe correspondence to these less symmetric situations in order to describe moregeneral phenomena in this context.One particular area of condensed matter that appears to be amenable to a holo-graphic description is the appearance of striped phases in certain materials.1 Thesephases are characterized by the spontaneous breaking of translational invariancein the system. Examples include charge density waves and spin density waves instrongly correlated electron systems, where either the charge and/or the spin densi-1Stripes are also known to form in large N QCD [30, 121].63ties become spatially modulated (for a review see [129]). The formation of stripesis conjectured to be related to the mechanism of superconductivity in the cuprates[19]. To approach this striking phenomenon from the holographic perspective, onewould look for an asymptotically ADS gravity system which allows a spontaneoustransition to a modulated phase.Recently, several interesting spatially modulated holographic systems have beenstudied. One way to study stripes on the boundary is to source them by imposingspatial modulation in the non-normalizable modes of some fields, explicitly break-ing the translation invariance, as in [49, 77].2 However, if one wishes to makecontact with the context described above, it is important that the inhomogeneityemerges spontaneously rather than be introduced explicitly.In some cases, the spatially modulated phase has an extra symmetry, allowingthe situation to be posed as a co-homogeneity one problem on the gravity side. Ex-amples include systems in which one of the translational Killing vectors is replacedby a helical Killing vector [33, 35, 36, 104–106]. More general inhomogeneous in-stabilities, in which one of the translation symmetries is fully broken, have beendescribed in a phenomenological model [34] and in certain #ND= 6 brane systems[14, 82, 83].3In this work, we study the full non-linear co-homogeneity two striped solu-tions to the Einstein-Maxwell-axion model that stem from the normalizable, in-homogeneous modes of the RN solution detailed in [34]. In this model, below acritical temperature, stripes spontaneously form in the bulk and on the boundary.We study the properties of the stripes in both the fixed length system, in which thewavenumber is set by the size of the domain and charges are integrated over thestripe, and the infinite system, in which the corresponding thermodynamic densi-ties are studied. For the black hole at fixed length, we examine the behavior indifferent thermodynamic ensembles as we vary the temperature and wavenumber.The study is facilitated by a numerical solution to the set of coupled Einsteinand matter equations in the bulk. Inspired by the black string case [124, 131], wefix the metric in the conformal gauge, resulting in a set of field equations and a2In a similar vein, more recently, lattice-deformed black branes have been of interest in studiesof conductivity in holographic models [42, 71–73].3Other studies of inhomogeneity in the context of holography include [11, 18, 79, 114].64set of constraint equations. Then, as described in [131], the resulting constraintequations can be solved by imposing particular boundary conditions on the fields.As well as being of interest from the holographic perspective these numericalsolutions are important as they represent new inhomogeneous black hole solutionsin Ads. We find strong evidence that the unstable homogeneous branes transitionsmoothly to the striped state below the critical temperature.4 As we approach zerotemperature the relative inhomogeneity is seen to grow without bound and the blackhole horizon tends to pinch off, signalling the formation of a spacetime singularityin this limit.A subset of our results has already been announced in [117], in this paper weprovide full details. The summary of the results follow:Boundary field theory• We calculate the fully back-reacted normalizable inhomogeneous modes.• The stripes have momentum, electric current and modulations in charge andmass density (see [93] for a recent study of angular momentum generation).• As a function of temperature, the modulations start small, then grow andsaturate as T → 0.• We study the stripe of fixed length in various ensembles, finding a secondorder phase transition, for sufficiently large axion coupling, in each of thegrand canonical (temperature T , chemical potential µ fixed), canonical (T ,charge N fixed) and microcanonical (mass M, N fixed) ensembles. We com-pute corresponding critical exponents.• For the infinite length system, there is a second order transition to a stripedphase. The width of the dominant stripe grows as the temperature is de-creased.• In the zero temperature limit, within the accuracy of our numerics, the en-tropy appears to approach a non-zero value.4 The instability to the formation of the striped black branes resembles the black string instability[53] which is known to be of the second order for high enough dimensions [89, 123].65Bulk geometryThe new inhomogeneous black brane solutions that we find have peculiar features,including• The inhomogeneities are localized near the horizon, and die off asymptoti-cally following a power law decay.• The phenomena of vorticity, frame dragging and the magneto-electric effectsimilar to one produced by a near horizon topological insulator are observed.• The inhomogeneous black brane has a neck and a bulge. In the curvature atthe horizon, the maximum is at the bulge. In the limit of small temperatures,the neck shrinks to zero size.• The proper length of the horizon grows when temperature is decreasing, anddiverges as 1/T 0.1 in the limit T → 0. The proper length in the stripe direc-tion increases from the boundary to the horizon, which can be thought of asa manifestation of an “Archimedes effect”.In §4.2, we define our model and set up our numerical approach, describingour ansatz, boundary conditions and solving procedure. Then, in §4.3, we reporton interesting geometrical features of the bulk solutions. §4.4 studies the solutionsat fixed length from the point of view of the boundary theory. There, we makethe comparison to the homogeneous solution and find a second order transition, inaddition to describing the observables in the theory. In §4.5, we relax the fixedlength condition and find the striped solution that dominates the thermodynamicsfor the infinite system. Appendix B.1 provides details about computing the observ-ables of the inhomogeneous solutions while appendix B.2 gives more details onthe numerics, including checks of the solutions and validations of our numericalmethod.Note added: As this manuscript was being completed, [32] and [132, 133]appeared, which use a different method and have some overlap with this work.664.2 Numerical Setup: Einstein-Maxwell-Axion ModelIn [34], perturbative instabilities of the RN black brane were found within theEinstein-Maxwell-axion model. In [117] and here, we construct the full non-linearbranch of stationary solutions following this zero mode.4.2.1 The model and ansatzThe Lagrangian describing our coupled system can be written as [34]L =12(R+12)− 12∂ µψ∂µψ− 12m2ψ2− 14FµνFµν− 1√−gc116√3ψ εµνρσFµνFρσ ,(4.2.1)where R is the Ricci scalar, Fµν is the Faraday tensor, ψ is a pseudo-scalar field andg is the determinant of the metric. We use units in which the ADS radius l2 = 1/2,Newton’s constant 8piGN = 1 and c = h¯ = 1, and choose m2 = −4. The constantc1 controls the strength of the axion coupling.For this choice of scalar field mass, instabilities exist for all choices of c1. Forc1 = 0, the instability is towards a black hole with neutral scalar hair. For c1 > 0,inhomogeneous instabilities along one field theory direction exist for a range ofwavenumbers k. The critical temperature at which each mode becomes unstabledepends on the wavenumber: Tc(k). For a given c1, there is a maximum criticaltemperature, above which there are no unstable modes. As one increases c1, thecritical temperature of a given mode k increases, such that for a fixed temperature alarger range of wavenumbers will be unstable. See appendix B.2.1 for more detailson the perturbative analysis.One may consider generalizations of this action, including higher order cou-plings between the scalar field and the gauge field. In particular, as discussed in[34], generalizing the Maxwell term as − τ(ψ)4 FµνFµν , where τ(ψ) is a functionof the scalar field, results in a model that can be uplifted to a D = 11 supergrav-ity solution (for particular choices of c1, m, and the parameters in τ(ψ)). In thisstudy, we wish to study the formation of holographic stripes phenomenologically.The existence of the axion-coupling term (c1 6= 0) is a sufficient condition for theinhomogeneous solutions and so we set τ(ψ) = 1 here.We are looking for stationary black hole solutions that can be described by an67ansatz of the formds2 =−2r2 f (r)e2A(r,x)dt2+e2B(r,x)(dr22r2 f (r)+2r2dx2)+2r2e2C(r,x)(dy−W (r,x)dt)2,ψ = ψ(r,x), A = At(r,x)dt+Ay(r,x)dy, (4.2.2)where r is the radial direction in ADS and x is the field theory direction along whichinhomogeneities form. We term the scalar field and gauge fields collectively as thematter fields. f (r) is a given function whose zero defines the black brane horizon.We take f (r) to be that of the RN solution,f (r) = 1−(1+µ24r20)(r0r)3+µ24r20(r0r)4, (4.2.3)so that the horizon is located at r = r0. The homogeneous solution is the RN blackbrane, given byA = B =C =W = ψ = Ay = 0, At(r) = µ(1− r0/r), (4.2.4)where µ is the chemical potential. Above the maximum critical temperature, thisis the only solution to the system.To find the non-linear inhomogeneous solutions, we numerically solve theequations of motion derived from the ansatz (4.2.2). The Einstein equation resultsin four second order elliptic equations, formed from combinations of Gtt −T tt = 0,Gty−T ty = 0, Gyy−T yy = 0, and Grr +Gxx− (T rr +T xx ) = 0, and two hyperbolic con-straint equations, Grx−T rx = 0 and Grr−Gxx− (T rr −T xx ) = 0, for the metric func-tions. The gauge field equations and scalar field equation give second order ellipticequations for the matter fields. For completeness, the full equations are given inappendix B.2.2. Our strategy will be to solve these seven elliptic equations subjectto boundary conditions that ensure that the constraint equations will be satisfied ona solution. Below, we describe the constraint system and our boundary conditions.For more details about the numerical approach, we refer to appendix B.2.684.2.2 The constraintsThe two equations Grx− T rx = 0 and Grr −Gxx− (T rr − T xx ) = 0, which we do notexplicitly solve, are the constraint equations. Using the Bianchi identities [131],we see that the constraints satisfy∂x(√−g(Grx−T rx ))+2r2√ f∂r(r2√ f√−g(Grr−Gxx− (T rr −T xx )))= 0,(4.2.5)2r2√f∂r(√−g(Grx−T rx ))−∂x(r2√ f√−g(Grr−Gxx− (T rr −T xx )))= 0.(4.2.6)Defining rˆ by ∂rˆ = 2r2√f∂r gives Cauchy-Riemann relations∂x(√−g(Grx−T rx ))+∂rˆ(r2√ f√−g(Grr−Gxx− (T rr −T xx )))= 0, (4.2.7)∂rˆ(√−g(Grx−T rx ))−∂x(r2√ f√−g(Grr−Gxx− (T rr −T xx )))= 0, (4.2.8)showing that the weighted constraints satisfy Laplace equations. Then, satisfyingone constraint on the entire boundary and the other at one point on the boundaryimplies that they will both vanish on the entire domain. In practice we will take ei-ther zero data or Neumann boundary conditions at the boundaries in the x-direction.The unique solution to Laplace’s equation with zero data on the horizon and theboundary at infinity and these conditions in the x-direction is zero. Therefore,as long as we fulfill one constraint at the horizon and the asymptotic boundaryand the other at one point (on the horizon or boundary), the constraints will besatisfied if the elliptic equations are. Our boundary conditions will be such that√−g(Grx−T rx ) = 0 at the horizon and conformal infinity and that r2√f√−g(Grr−Gxx− (T rr −T xx )) = 0 at one point on the horizon.696-regularity,√−gGrx = 0A,B,C,W ∝ 1r3 ,At −µ,Ay ∝ 1r ,ψ ∝ 1r2 ,√−gGrx = 0ψ = ∂xAy = ∂xgty = ∂xh = 0∂xψ = Ay = gty = ∂xh = 0rr = rcutr = r0xx = L4x = 0Figure 4.1: A summary of the boundary conditions on our domain. At thehorizon, r = r0, we impose regularity conditions. At the conformalboundary, r → ∞, we have fall off conditions on the fields (imposedat large but finite r = rcut) such that we do not source the inhomo-geneity. In the x-direction, we use symmetries to reduce the domainto a quarter period L/4. Then, we impose either periodic or zero con-ditions on the fields, according to their behavior under the discretesymmetries discussed in the text. (h collectively denotes the fields{gtt ,gxx,gyy,At}.) In addition to these, we explicitly satisfy the con-straint equation√−gGrx = 0 on the horizon and the conformal bound-ary.4.2.3 Boundary conditionsThe elliptic equations to be solved are subject to physical boundary conditions.There are four boundaries of our domain (see Fig. 4.1): the horizon, the conformalboundary, and the periodic boundaries in the x-direction, which are described next.Staggered periodicityTo specify the boundary conditions in the x direction we look at the form of thelinearized perturbation which becomes unstable (see appendix B.2.1). To leading70order in the perturbation parameter λ , they are of the form:ψ(x) ∼ λ cos(kx),Ay(x) ∼ λ sin(kx),gty(x) ∼ λ sin(kx), (4.2.9)where k is the wavenumber of the unstable mode. To second order in the perturba-tion parameter, the functions gtt ,gxx,gyy and At (which we denote collectively as h)are turned on, with the schematic behaviorh(x)∼ λ 2(cos(2kx)+C), (4.2.10)where C are independent of x.All these functions are periodic with period L = 2pik . However, they are notthe most general periodic functions with period L. For numerical stability it isworthwhile to specify their properties further and encode those properties in theboundary conditions we impose on the full solution. We concentrate on the behav-ior of the perturbation with respect to two independent Z2 reflection symmetries.The first Z2 symmetry is that of x→−x, y→−y, which is a rotation in the x,yplane. This is a symmetry of the action and of the linearized perturbation (keepingin mind that Ay and gty change sign under reflection of the y coordinate). Weconclude therefore that this is a symmetry of the full solution.Similarly, the Z2 operation x→ L2 − x, y→ −y is a symmetry of the action,which is also a symmetry of the linearized system when accompanied by λ →−λ .In other words the functions ψ,Ay,gty are restricted to be odd with respect to thisZ2 operation, while the rest of the functions, which we collectively denoted as h,are even.The two symmetries defined here restrict the form of the functions that canappear in the perturbative expansions for each of the functions above. For example,it is easy to see that the function ψ(x) gets corrected only in odd powers of λ andthe most general form of the harmonic that can appear in the perturbative expansionis cos(nkx), for n odd. Similar comments apply to the other functions above.We restrict ourselves to those harmonics which may appear in the full solution.71The most efficient way to do so is to work with a quarter of the full period L(reconstructing the full periodic solution using the known behavior of each functionwith respect to the two Z2 operations defined above). The specific properties ofeach function appearing in our solutions are imposed by demanding the followingboundary conditions:∂xψ(x = 0) = 0, ψ(x =L4) = 0,Ay(x = 0) = 0, ∂xAy(x =L4) = 0,gty(x = 0) = 0, ∂xgty(x =L4) = 0,∂xh(x = 0) = 0, ∂xh(x =L4) = 0. (4.2.11)At the horizonIn our coordinates (4.2.2) the horizon is at fixed r = r0. For numerical conveniencewe introduce another radial coordinate ρ =√r2− r20, such that the horizon is atρ = 0.5 Expanding the equations of motion around ρ = 0 yields a set of Neumannregularity conditions,∂ρA = ∂ρC = ∂ρW = ∂ρψ = ∂ρAt = ∂ρAy = 0, (4.2.12)and two conditions in the inhomogeneous direction along the horizon,∂xW = ∂x(At +WAy) = 0. (4.2.13)Thus, both W and the combination At +WAy are constant along the horizon. Theboundary conditions in the x direction (4.2.11) imply that W = 0. Then, the secondcondition together with regularity of the vector field A on the Euclidean sectiongive that At = 0 on the horizon.The regularity conditions give eight conditions for the six functions A,C,W,ψ,Atand Ay. In principle, we would choose any six of these to impose at the horizon.5In the rest of the paper, we use r and ρ interchangeably as our radial coordinate. We use thecoordinate ρ in the numerics.72If we find a non-singular solution to the equations, then the other two conditionsshould also be satisfied. In practice, some of these conditions work better thanothers for finding the numerical solution. We find that using Neumann conditionsfor A,C,ψ , and Ay and Dirichlet conditions for W and At results in a more stablerelaxation.6The conditions for B are determined using the constraint equations. Expandingthe weighted constraints at the horizon, we find√−g(Grx−T rx ) ∝ ∂x(A−B)+O(ρ), (4.2.14)r2√f√−g(Grr−Gxx− (T rr −T xx )) ∝ ∂ρB+O(ρ). (4.2.15)The first condition gives constant surface gravity (or temperature) along the hori-zon. As discussed above, we will impose one constraint at the horizon and theboundary, and the other at one point. In practice, we will satisfy r2√f√−g(Grr−Gxx− (T rr −T xx )) at (ρ,x) = (0,0), updating the value of B at this point using theNeumann condition ∂ρB = 0. This will set the difference (B−A)|(ρ,x)=(0,0) ≡ d0,which we will then use to update B using a Dirichlet condition along the rest of thehorizon, satisfying√−g(Grx−T rx ) = 0.At the conformal boundaryIn our coordinates, the boundary is at r =∞. Since we are looking for spontaneousbreaking of homogeneities, our boundary conditions will be such that the fieldtheory sources are homogeneous. This implies that the non-normalizable modesof the bulk fields are homogeneous. The inhomogeneity of the striped solutionswill be imprinted on the normalizable modes of the fields, or the coefficient of thenext-to-leading fall-off term in the asymptotic expansions.The form of our metric ansatz is such that the metric functions A,B,C and Wrepresent the normalizable modes of the metric. Imposing that the geometry isasymptotically ADS with Minkowski space on the boundary implies that these fourmetric perturbations must vanish as r→ ∞. By expanding the equations of motionnear the boundary, one can show that A,B,C and W fall off as 1/r3. In practice,6Using Neumann conditions at the horizon for W and At results in values at the horizon thatconverge to zero with step-size, consistent with the above analysis.73we place the outer boundary of our domain at large but finite rcut and impose thefall-off conditions there.As in the RN solution, we source the field theory charge density with a homo-geneous chemical potential, corresponding to a Dirichlet condition for the gaugefield At at the boundary. In the inhomogeneous solutions, we expect the sponta-neous generation of a modulated field theory current jy(x), dual to the normalizablemode of Ay. Solving the equations near the boundary with these conditions revealsthe expansions At = µ +O(1/r) and Ay = O(1/r), which we impose numericallyat rcut .The scalar field equation of motion gives the asymptotic solutionψ =ψ(1)rλ−+ψ(2)rλ++ . . . , (4.2.16)whereλ± =12(3±√9+4(lm)2). (4.2.17)For the range of scalar field masses−9/2≤m2 ≤−5/2, both modes are normaliz-able, and fixing one mode gives a source for the other. In our study we will choosem2 =−4, giving λ− = 1, λ+ = 2. Since we are looking for spontaneous symmetrybreaking, in this case we must choose either ψ(1) = 0 or ψ(2) = 0. We choose theformer, so that ψ falls off as 1/r2.Now, consider the weighted constraint√−gGrx. As discussed above, in order tosolve the constraint system, we require this to disappear at the conformal boundary.Near the boundary,√−g ∝ r2 + . . . , so for √−gGrx to disappear we must haveGrx = O(1/r3). Expanding the equations near the boundary we haveGrx−T rx ∝3∂xA(3)(x)+2∂xB(3)(x)+3∂xC(3)(x)r2+O(1r3), (4.2.18)where X = X (3)(x)/r3+ . . . for X = {A,B,C}. Therefore, in addition to the bound-ary conditions mentioned above, for√−gGrx = 0 to be satisfied at r =∞, it appearsthat we should have that 3A(3)(x)+2B(3)(x)+3C(3)(x) = const. The means to im-pose this addition condition comes from the fact that our metric (4.2.2) has an un-fixed residual gauge freedom [3], allowing one to transform to new r˜ = r˜(r,x), x˜ =74x˜(r,x) coordinates which are harmonic functions of r and x. Performing such atransformation generates an additional function in (4.2.18), which can then be cho-sen to ensure that the constraint is satisfied (in appendix B.2 we describe how).This condition implies the conservation of the boundary energy momentum tensor,see appendix B.1.4.2.4 Parameters and algorithmThe physical data specifying each solution is the chemical potential µ , the tem-perature T , and the periodicity L.7 Since the boundary theory is conformal, it willonly depend on dimensionless ratios of these parameters. This manifests itself inthe following scaling symmetry of the equations:r→ λ r, (t,x,y)→ 1λ(t,x,y), Aµ → λAµ . (4.2.19)We use this to select µ = 1. Then, our results are functions of the dimensionlesstemperature T/µ and the dimensionless periodicity Lµ .The temperature is controlled by the coordinate location of the horizon. Fora given r0, the temperature of the RN phase is T0 = (1/8pir0)(12r20− 1) while thetemperature of the inhomogeneous solution is T = e−d0T0. Recall that (B−A)|r0 =d0 is dynamically generated by satisfying the constraints at the horizon. Fromour numerical solutions, we find that d0 monotonically increases as we lower thetemperature, so that T0 gives a reliable parametrization of the physical temperatureT . In practice, we generate solutions by choosing values of T0 below the criticaltemperature Tc(k).We solve the equations by finite-difference approximation (FDA) techniques.We use second order FDA on the equations (B.2.6) - (B.2.12) before using a point-wise Gauss-Seidel relaxation method on the resulting algebraic equations. For theresults in this paper, for c1 = 4.5, a cutoff of ρcut = {6,8} was used while forc1 = 5.5 and c1 = 8, for which the modulations were larger, a cutoffs of ρcut =10 and ρcut = 12 correspondingly were used. Grid spacings used for the FDA7Fixing µ , T and L gives the system in the grand canonical ensemble. Once the phase space hasbeen mapped in one ensemble other ensembles can be considered via appropriate reinterpretation ofthe numerical data. See §4.4 for a description of this process.75scheme were in the range dρ,dx = 0.04− 0.005. Neumann boundary conditionsare differenced to second order using one-sided FDA stencils in order to updatethe boundary values at each step. At the asymptotic boundary ρcut we impose theboundary conditions by second order differencing a differential equation based onthe fall-off (for example, ∂rA = −3A/r) to obtain an update rule for the boundaryvalue. As a result we find quadratic convergence as a function of grid-spacing forour method, see appendix B.2.5.4.3 The SolutionsThe system of equations (B.2.6-B.2.12) is solved subject to boundary conditiondescribed in the previous sections. The details of our numerical algorithm arefound in appendix B.2. Here we focus on the properties of the solutions and theirgeometry.Unless otherwise specified the following plots were obtained using the axioncoupling of c1 = 4.5. In this section, we consider solutions for which the periodic-ity is determined by the dominant critical wavenumber kc; for c1 = 4.5, this givesLµ/4 ' 2.08, see Table B.1. We found that the geometry and most of the otherfeatures are qualitatively similar for the couplings c1 = 5.5 and c1 = 8. A con-venient way to parametrize our inhomogeneous solutions is by the dimensionlesstemperature T/Tc, relative to the critical temperature Tc, below which the transla-tion invariance along x is broken. For c1 = 4.5, our method allows us find solutionsin the range 0.003. T/Tc . 0.9.4.3.1 Metric and fieldsFor subcritical temperatures, as we descend into inhomogeneous regime, the metricand the matter fields start developing increasing variation in x. Fig. 4.2 displays themetric functions, and Fig. 4.3 shows the non vanishing components of the vectorpotential field and of the scalar field for T/Tc ' 0.11 over a full period in the xdirection. The variation of all fields is maximal near the horizon of the black holeat ρ =√r2− r20 = 0, and it gradually decreases toward the conformal boundary,ρ → ∞.Many of the special features of the solutions we find may be explained via76Figure 4.2: Metric functions for T/Tc ' 0.11. Note the metric functions A,Band C have half the period of W . The variation is maximal near thehorizon, located at ρ = 0, and it decays as the conformal boundary isapproached, when ρ → ∞.axion electrodynamics as seen in the effective description of the electromagneticresponse of a topological insulator. This effect is mediated by the interaction termin our Lagrangian (4.2.1). In the broken phase we have an axion gradient in the nearhorizon geometry, which realizes a topological insulator interface, see Fig. 4.3.The characteristic patterning of the near horizon magnetic field, B=∇×A, shownin Fig. 4.4, is reminiscent of the magnetoelectric effect at such interfaces. Themagnetic vortices are localized near the black hole horizon and have alternatingdirection of magnetic field lines.In curved space the magnetic field is accompanied by vorticity, which is man-ifested by the function W . This causes frame dragging effects in the y direction.77Figure 4.3: At relative to the corresponding RN solutions, Ay and ψ forT/Tc ' 0.11. The period of At is twice that of ψ and Ay. The x-dependence dies off gradually as the conformal boundary is approached,at ρ → ∞.Test particles will be pushed along y with speeds W (r,x), in particular the directionof the flow reverses every half the period along x. The drag vanishes at the horizonand at the location of the nodes of W where x = nL/2, for integer n, see Fig. 4.2.In general, the dragging effect remains bounded, and no ergoregion forms, wherethe vector ∂t becomes spacelike.4.3.2 The geometryThere are several ways to envisage the geometry of our solutions, we discuss themin turn.The Ricci scalar of the RN solution is RRN =−24, constant in r and independentof the parameters of the black hole. This is no longer true for the inhomogeneous78−4 −3 −2 −1 0 1 2 3 400.20.40.60.81ρxFigure 4.4: Magnetic field lines for solution with T/Tc ' 0.07. The patternof vortices of alternating field directions form at the horizon (located atρ = 0).phases, where the Ricci scalar becomes position dependent. Fig. 4.5 illustratesthe spatial variation of the Ricci scalar, relative to the RRN for T/Tc ' 0.054. Themaximal curvature is always along the horizon at x = nL/2 for integer n. It growswhen the temperature decreases and approaches the finite value of R'−94 in thesmall temperature limit.Embedding in a given background space is a convenient way to illustrate curvedgeometry. We consider the embedding of 2-dimensional spatial slices of constantx of the full geometry (4.2.2)ds22 =e2B(r,x)2r2 f (r)dr2+2r2 e2C(r,x)dy2 (4.3.1)as a surface in 3-dimensional ADS spaceds23 = 2 r˜2 dz2+dr˜22 r˜2+2 r˜2dy2. (4.3.2)We are looking for a hypersurface parametrized by z = z(r˜). Then the metric on79Figure 4.5: Ricci scalar relative to that of RN black hole, R/RRN −1, RRN =−24, for T/Tc ' 0.054 over half the period. The scalar curvature ismaximal along the horizon at x = nL/2 for integer n.such a hypersurface readsds22 =[1+2 r˜2(dzdr˜)2] dr˜22 r˜2+2 r˜2dy2. (4.3.3)Comparing (4.3.3) and (4.3.1) we obtain set of the relationsr˜ = r eC,[12 r˜2+2 r˜2(dzdr˜)2](dr˜dr)2=e2B(r,x)2r2 f (r), (4.3.4)80Figure 4.6: The embedding diagram of constant x spatial slices, as a functionof x at given y for T/Tc ' 0.035. The geometry of ρ = const slices ismaximally curved at x = nL/2 for integer n.resulting in the embedding equationdzdr=12r2√f (r)−1 e2B(r,x)−2C(r,x)− (1+ r∂rC(r,x))2. (4.3.5)We integrate this equation for a given x, and in Fig. 4.6 show the embedding atconstant y. The maximal curvature along ρ = const slices occurs at x = nL/2 forinteger n, which is consistent with Fig. 4.5.The proper length of the stripe along x relative to the background ADS space-time at given r islx(r)/lx(r = ∞) =∫ L/40eB(r,x)dx. (4.3.6)Fig. 4.7 shows the dependence of the normalized proper length on the radial dis-tance from the horizon. The proper length tends to the coordinate length as 1/r3asymptotically as r→∞, but it exceeds that as the horizon is approached. Namely,the inhomogeneous black brane “pushes space” around it along x, in a mannerresembling the “Archimedes effect”.8110−2 10−1 10000.10.20.30.40.5log(ρ)l x(ρ)/l x(∞)−1Figure 4.7: Radial dependence of the normalized proper length along x forT/Tc ' 0.054. While asymptotically the proper length coincides withthe coordinate size of the strip, it grows as the horizon is approached.This is a manifestation of the “Archimedes effect”.The proper length of the horizon in x direction is obtained calculating (4.3.6) atr0. Fig. 4.8 demonstrates the dependence of this quantity on the temperature. Forhigh temperatures the length of the horizon resembles that of the homogeneousRN solution, however, it grows when temperature decreases. We find that at smallT/Tc the proper length of the horizon diverges approximately as (T/Tc)−0.1.The transverse extent of the horizon, per unit coordinate length y, is given byry(x) =√2r0 eC(r0,x). (4.3.7)Fig. 4.9 shows the variation of ry(x) along the horizon for T/Tc ' 0.054. Typicallythere is a “bulge” occurring at x = nL/2 and a “neck” at x = (2n+ 1)L/4, forinteger n. Comparing this with Fig. 4.5 we note that Ricci scalar curvature ismaximal at the bulge and not at the neck as would happen, for instance, in thecylindrical geometry in black string case [124]. Fig. 4.10 displays the dependenceof the sizes of the neck and bulge on T/Tc. Both sizes monotonically decreasewith temperature, however the rate at which the neck is shrinking exceeds that of820 0.1 0.2 0.3 0.411.11.21.31.41.51.61.71.81.9T /TcLh/LRNh  datafit to Lh/LhRN ∼ (T/Tc)−0.1Figure 4.8: Temperature dependence of the proper length of the horizonalong the stripe. Starting from as low as L at high temperatures, theproper length grows monotonically and for small T/Tc the growth iswell approximated by the power-law dependence ∼ (T/Tc)−0.1.−2 −1 0 1 200.050.1r yxFigure 4.9: The extent of the horizon in the transverse direction, ry, as a func-tion of x for T/Tc ' 0.054 in x∈ [−L/2,L/2]. The characteristic patternof alternating “necks” and “bulges” forms along x.the bulge. This is demonstrated in Fig. 4.11. In fact, we find that for c1 = 4.5,rnecky /rbulgey ∼ (T/Tc)1/2 near the lower end of the range of temperatures that weinvestigated. For other values of the axion coupling the scaling of the ratio is againpower-law, with an exponent of the same order of magnitude, e.g. for c1 = 8, theexponent is about 0.12. This signals a pinch-off of the horizon in the limit T → 0.83−2.5 −2 −1.5 −1 −0.50.050.10.150.20.250.30.350.4rylog(T/Tc)  neckbulge homogeneous RNFigure 4.10: The dependence of the size of the neck and the bulge on tem-perature.10−3 10−2 10−1 10010−1100T /Tcrnecky/rbulgey  datafit to ryneck/rybulge  = 2 (T/Tc)0.5Figure 4.11: The ratio of the transverse extents of the neck and the bulgeshrinks as rnecky /rbulgey ∼ (T/Tc)1/2 at small temperatures, indicating apinch-off of the horizon in the limit T → 0.844.4 Thermodynamics at Finite LengthIn this section we consider the thermodynamics and phase transitions in the system,assuming that the stripe length is kept fixed. For the finite system the length ofthe interval is part of the specification of the ensemble and is kept fixed. In thenext section we discuss the infinite system, for which the stripe width can adjustdynamically.4.4.1 The first lawWe demonstrated that below the critical temperature there exists a new branch ofsolutions which are spatially inhomogeneous. In the microcanonical ensemble thecontrol variables of the field theory are the entropy S, the charge density N, andthe length of the x-direction L, with corresponding conjugate variables temperatureT , chemical potential µ , and tension in the x-direction τx8. The usual first lawis augmented by a term corresponding to expansions and contractions in the x-direction and is given bydM = T dS+µdN+ τxdL. (4.4.1)where M, S, and N are quantities per unit length in the trivial y direction, but areintegrated over the stripe.Our system has a scaling symmetry given by (4.2.19). In the field theory, thiscorresponds to a change of energy scale. Under this transformation, the thermody-namic quantities scale asM→ λ 2M, T → λT, S→ λS, µ → λµ,N→ λN, τx→ λ 3τx, L→ 1λ L. (4.4.2)Using these in (4.4.1) with λ = 1+ ε , for ε small, yields2M = T S+µN− τxL, (4.4.3)the Smarr’s-like expression that our solutions must satisfy and that can be used as8Explicit expressions for these quantities in terms of our ansatz are given in appendix B.1.85a check of our numerics. For all of our solutions, we have verified that this identityis satisfied to one percent.4.4.2 Phase transitionsThe question of which solution dominates the thermodynamics depends on theensemble considered. In the holographic context the choice of thermodynamic en-semble is expressed through the choice of boundary conditions. The correspondingthermodynamic potential is computed as the on-shell bulk action, appropriatelyrenormalized and with boundary terms rendering the variational problem well-defined. We examine each ensemble in turn.The grand canonical ensembleIn our numerical approach, the natural ensemble to consider is the grand canonicalensemble, fixing the temperature T , the chemical potential µ , and the periodicityof the asymptotic x direction as L. The corresponding thermodynamic potential isthe grand free energy densityΩ(T,µ,L) = M−T S−µN. (4.4.4)Different solutions of the bulk equations with the same values of T,µ,L correspondto different saddle point contributions to the partition function. The solution withsmallest grand free energy Ω is the dominant configuration, determining the ther-modynamics in the fixed T,µ,L ensemble. In our case we have two solutions foreach choice of T,µ,L, one homogeneous and one striped. Exactly how one onesaddle point comes to dominate over the other at temperatures below the criticaltemperature determines the order of the phase transition.In this ensemble it is convenient to measure all quantities in units of the fixedchemical potential µ . Then, after fixing L from the critical mode appearing at thehighest Tc (see Fig. B.1 and Table B.1 in appendix B.2.1), we have that Ω/µ2 is afunction only of the dimensionless temperature T/µ . In the fixed chemical poten-tial ensemble for large enough axion coupling we find a second order transition,where the inhomogeneous charge distribution starts dominating the thermodynam-ics immediately below the temperature at which the inhomogeneous instability de-86L̐4=1.400.0 0.2 0.4 0.6 0.8 1.0-0.05-0.04-0.03-0.02-0.010.00TTcHW-WRNLΜ2L̐4=1.210.0 0.2 0.4 0.6 0.8 1.0-0.04-0.03-0.02-0.010.00TTcHW-WRNLΜ2L̐4=1.040.0 0.2 0.4 0.6 0.8 1.0-0.020-0.015-0.010-0.0050.000TTcHW-WRNLΜ2 L̐4=0.920.0 0.2 0.4 0.6 0.8 1.0-0.015-0.010-0.0050.000TTcHW-WRNLΜ2Figure 4.12: The grand free energy relative to the RN solution for severalsolutions of different fixed lengths at c1 = 8. In all cases shown weobserve a second order phase transition. The critical exponents de-termined near the critical points in each case are consistent with thequadratic behavior (Ω−ΩRN)/µ2 ∝ (1−T/Tc)2.velops. Near the critical temperature, the behavior of the grand free energy differ-ence is consistent with (Ω−ΩRN)/µ2 ∝ (1−T/Tc)2, while the entropy differencegoes as (S− SRN)/µ ∝ T/Tc− 1. This is as expected from a second order transi-tion. As can be seen in Fig. 4.12 and Fig. 4.13, we find this second order transitionfor a range of lengths, L, and for a variety of values of the axion coupling c1. Withthe current accuracy of our numerical procedure, we find it increasingly difficultto resolve the order of the phase transition for smaller values of c1. In fact, forc1 = 4.5 the grand free energies of the homogeneous and inhomogeneous phasesare nearly degenerate but still allow us to determine the phase transition as secondorder. It would be interesting to see if the phase transition remains of second orderor changes to the first order for smaller values of the axion coupling.To examine the observables in the striped phase further, we focus on c1 = 8 andthe corresponding dominant critical mode, Lµ/4' 1.21, and consider solutions for870.0 0.2 0.4 0.6 0.8 1.0-0.008-0.006-0.004-0.0020.000TTcHW-WRNLΜ2Figure 4.13: The grand free energy relative to the RN solution for c1 = 4.5and fixed Lµ/4 = 2.08. The grand free energies of the homogeneousand inhomogeneous phases are nearly degenerate, such that their max-imal fractional difference is about 1%.the temperatures 0.00016 . T/Tc . 0.96. Various quantities are plotted with thecorresponding homogeneous results in Fig. 4.14. Along this branch of solutions,the mass of the stripes is more than the RN solution and the entropy is always less.We plot the maximum of the boundary current density 〈 jy〉, momentum density〈Ty0〉 and pseudoscalar operator vev 〈Oψ〉. Fitting the data near the critical point tothe function (1−T/Tc)α , we find the approximate critical exponents α jy = 0.40,αTy0 = 0.41 and αOψ = 0.38 with relative fitting error of about 10%.We find evidence that the entropy of the striped black branes does not tend tozero in the small temperature limit, see Fig. 4.14. This is further supported bythe behavior of the transverse size of the horizon (4.3.7). Here the bulge seemsto contract at a much slower rate than the neck, which evidently shrinks to zerosize in the limit T → 0. However, strictly speaking, this conclusion is based onextrapolation of the finite temperature data to T = 0. Checking whether the entropyasymptotes to a finite value or goes to zero in this limit, as suggested in [132, 133],will require further investigation with a method of higher numerical accuracy.880.0 0.2 0.4 0.6 0.8 1.00.40.50.60.70.80.9TTcMΜ20.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.5TTcSΜ0.0 0.2 0.4 0.6 0.8 1.00.70.80.91.0TTcNΜ0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.100.12TTc<j y>maxΜ0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08TTc<OΨ>maxΜ0.0 0.2 0.4 0.6 0.8 1.00.000.050.100.150.20TTc<T y0>maxΜ2Figure 4.14: The observables in the grand canonical ensemble for c1 = 8 andLµ/4 = 1.21 (points with dotted line) plotted with the correspondingquantities for the RN black hole (solid line). Fitting the data near thecritical point to the function (1− T/Tc)α , we find the approximatecritical exponents α jy = 0.40, αTy0 = 0.41 and αOψ = 0.38 with relativefitting error of about 10%.89The canonical ensembleTo study the system in the canonical ensemble we fix the temperature, total chargeand length of the system. This describes the physical situation in which the sys-tem is immersed in a heat bath consisting of uncharged particles. The relevantthermodynamic potential in this ensemble is the free energy densityF(T,N,L) = M−T S. (4.4.5)If we measure all quantities in units of the fixed charge N, then, again, the freeenergy F/N2 is only a function of the dimensionless temperature T/N.To solve our system with a fixed charge, we would need to fix the integral in xof the coefficient of the 1/r term in the asymptotic expansion of the gauge field At .Numerically, it is much easier to fix the chemical potential, as this gives a Dirich-let condition on At at the boundary. In the grand canonical ensemble, we solvedfor one-parameter families of solutions at fixed Lµ , labelled by the dimensionlesstemperature T/µ . Equivalently, in the (Lµ,T/µ) plane, we solve along the line offixed Lµ . Translated to the situation in which we measure quantities in terms ofthe charge density N, these solutions become one-parameter families of solutionswith varying LN, or a curve in the (LN,T/N) plane with LN a function of T/N.By varying the length Lµ (or solving with µ = 1 and varying L), we can find a col-lection of solutions that intersect the desired fixed LN line. By interpolating thesesolutions and evaluating the interpolants at fixed LN, we can study the stripes inthe canonical ensemble.In this ensemble we find a similar second order transition, in which the inho-mogeneous solution dominates the thermodynamics below the critical temperature(Fig. 4.15). The scaling of the relative free energy density slightly below the criti-cal temperature appears to fit a linear scaling, however using more points at lowertemperatures in the fit increases the critical exponent towards a quadratic scaling,as expected in a second order transition.The microcanonical ensembleThe microcanonical ensemble describes an isolated system in which all conservedcharges (in this case the mass and the charge) are fixed. This ensemble describes900.0 0.2 0.4 0.6 0.8 1.0-0.30-0.25-0.20-0.15-0.10-0.050.00TTcHF-F RNLN2Figure 4.15: The difference in canonical free energy, at c1 = 8 and fixedLN/4 = 1.25, between the striped solution and the RN black hole. Thestriped solution dominates immediately below the critical temperature,signalling a second order phase transition.the physical situation relevant to the study of the real time dynamics of an isolatedblack brane at fixed length. In this case, the state that maximizes entropy is thedominant solution. As shown in Fig. 4.16, we find that the entropy of our inhomo-geneous solutions is always greater than that of the RN black hole of the same mass.Furthermore, the mass of the inhomogeneous solutions is always smaller than thatof the critical RN black hole. Therefore, at fixed LN, the unstable RN black holesbelow critical temperature are expected to decay smoothly to our inhomogeneoussolution.Fixing the tensionAlternatively, one could attempt to compare solutions with different values of L.The meaningful comparison is in an ensemble fixing the tension τx. For example,91Extremal RN0.72 0.74 0.76 0.78 0.80 0.82 0.84MN20.00.51.01.52.02.5SNFigure 4.16: The entropy of the inhomogeneous solution for c1 = 8 (pointswith dotted line) and of the RN solution (solid line). Below the criticaltemperature, the striped solution has higher entropy than the RN. TheRN branch terminates at the extremal RN black hole, while the stripedsolution persists to smaller energies.one could compare the Legendre transformed grand free energyG(T,µ,τx) = M−T S−µN− τxL (4.4.6)where the additional terms comes from boundary terms in the action rendering thenew variational problem (fixing τx) well-defined. The candidate saddle points arethe solutions we find with various periodicities L, and their relative importance inthe thermodynamic limit is determined by G(T,µ,τx). In particular the solutionwhich is thermodynamically dominant depends on the value of τx we hold fixed.In this study we concentrate on the thermodynamics in the fixed L ensemble andwe leave the study of the fixed τx ensemble to future work.924.5 Thermodynamics for the Infinite SystemIn this section we lift the assumption of the finite extent of the system in the x-direction and consider the thermodynamics of the formation of the stripes belowthe critical temperature. For the infinite system we can define densities of thermo-dynamic quantities along x:m =ML, s =SL, n =NL. (4.5.1)In terms of these, the first law for the system becomesdm = T ds+µdn (4.5.2)and the conformal identity is3m = 2(T s+µn). (4.5.3)In the infinite system, we compare stripes of different lengths, at fixed T/µ ,to each other and to the homogeneous solution. The solution that dominates thethermodynamics is the one with the smallest free energy density ω , whereω = m−T s−µn. (4.5.4)This comparison is shown in Fig. 4.17 for c1 = 8, where we see that the free en-ergy density of the stripes is negative relative to the RN black hole, indicating thatthe striped phase is preferred at every temperature below the critical temperature.9Very close to the critical temperature, the dominant stripe is that with the criticalwavelength kc. As we lower the temperature, the minimum of the free energy den-sity traces out a curve in the (Lµ,T ) plane, and the dominant stripe width increasesto Lµ/4≈ 2.One can also study the observables of the system along this line of minimumfree energy density. The results are qualitatively similar to those for the fixed Lsystem (Fig. 4.14). In particular, the free energy density scales as (ω−ωRN)/µ3 ∝(1−T/Tc)2 near the critical point, indicating a second order transition in the infi-9In appendix B.2.4, we describe the generation of Fig. 4.17.931.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10L̐4TΜHΩ-ΩRNLΜ3-0.04-0.03-0.02-0.010.00Figure 4.17: Action density for c1 = 8 system relative to the RN solution. Thered line denotes the approximate line of minimum free energy.nite system as well.94Chapter 5Fermi Liquids from D-Branes5.1 Introduction and OutlineIn this chapter we explore the application of gauge-gravity duality to models ofFermi liquids. It has previously been shown that probe fermions in the RN blackhole or Lifshitz backgrounds exhibit features characteristic of a non-Fermi liquid[24, 92]. Much follow up effort was been devoted to the difficult tasks of intro-ducing quantum mechanical fermions into the bulk and to going beyond the probelimit. See in particular [5, 6, 119] for works most relevant to our current endeavour.While the above gravity duals utilize the bulk closed-string sector, many holo-graphic models also utilize open string sectors, i.e. probe D-BRANES [85] whichmay be embedded in an ambient space-time without back-reacting on the geome-try. In this work we use such D-brane constructions to study a new class of holo-graphic matter resulting from the inclusion of worldvolume fermions. In the spiritof bottom-up holography we consider a model which includes the minimal set ofingredients needed to construct the state we are interested in. Our bosonic fields arethen a gauge field and an embedding function, governed by the Dirac-Born-Infeld(DBI) action. These are accompanied by charged world-volume fermionic matter,which for the sake of simplicity we choose to be a massive Dirac fermion on theD-brane.The bulk solutions in our model consist of a compact (“Minkowski”) brane em-beddings whose gauge field and embedding function are coupled to a finite density95of charged fermions on the world-volume. Such compact embeddings are knownto be unstable if the charged matter is bosonic. This instability can be understoodas a result of Bose-Einstein condensation of the charged bosons at the point of thebrane cap off. This will manifest itself geometrically in the brane embedding be-ing pulled towards the interior of the geometry [88]. The new ingredient for usis Fermi statistics, resulting in an additional effective pressure, the Fermi pressureof the worldvolume Fermi surface. By constructing the state numerically, we ex-plicitly show that for sufficiently dense fermions, such pressure can stabilize thecompact brane embedding.The state in the dual field theory is shown to be a Fermi liquid, in that it has asharp Fermi surface (at zero temperature), and the low energy fermionic degrees offreedom have a quasiparticle description. The resulting Fermi-like liquid is similarto that constructed in [119]. We demonstrate the existence of a Fermi surface anddiscuss some qualitative features of the quasiparticle scattering rate. We also iden-tify limits of parameter space where perturbation theory is likely to break down,resulting in a qualitative change in the nature of the fermionic state. We there-fore propose that this state may provide a useful starting point for the constructionof non-Fermi liquids. Such a construction will need to tackle the difficult issuesaddressed in [5, 6] in the gravitational context.The layout of this paper is as follows. In Section 5.2 we introduce the bosonicand fermionic components of our action and derive the equations of motion andboundary conditions. We also take this opportunity to discuss the various param-eters and couplings in our phenomenological probe brane action. In Section 5.3we explain our numerical process, especially the unique features associated withimposing Fermi statistics, following the discussion of [119]. We then present oursolutions for the bulk fields, first in the probe limit and then including the backre-action of the fermions on the brane embedding. We focus especially on identifyinglimits where the perturbative expansion in 1N , utilized here, is likely to breakdown. In Section 5.4 we analyze the state of the dual QFT. We demonstrate theexistence of a Fermi surface via examination of the retarded Green’s function anddiscuss the equation of state. We conclude with some remarks on potential avenuesfor future research.965.2 Setup: Equations and Boundary ConditionsWe take a phenomenological, bottom-up approach to the problem of constructingfinite density fermionic states on probe D-BRANES. Thus, in constructing our fi-nite density state we do not commit to the matter content and full set of couplingsresulting from any specific brane configuration. Rather, we take the minimal set ofingredients necessary to construct the state we are interested in. Here we enumer-ate those basic ingredients needed for the construction. We comment below on theexpected impact of varying the matter content and couplings of our phenomeno-logical model.5.2.1 Bosonic sectorThe starting point for our phenomenological model is a Dp-Dq system in the holo-graphic decoupling limit: a single (or a few) Dq-probe branes in the near-horizongeometry of a stack of Dp-branes. Since we would like to study a 2+1 dimensionalQFT, we choose p = 2. The simplest type of probe brane is q = p+ 4, a systemthat has been extensively studied starting with [85]. We choose therefore to studythe worldvolume dynamics of a single D6 brane in the near horizon geometry ofa stack of D2-branes1. This brane configuration was discussed in the holographiccontext, e.g. in [86, 88, 101].This construction provides the basic elements needed to study holographic fi-nite density matter: a world volume gauge field which can be sourced by a chemi-cal potential, and an embedding function which can adjust to the presence of finitedensity matter. These are analogous to the metric and bulk gauge field in holo-graphic constructions utilizing the bulk closed string sector, the main differencebeing the DBI action controlling their couplings. Here we explore consequencesof these differences.Finite density holographic matter on D-BRANES, in the absence of chargedfermions, was discussed in [88]. In the presence of non-zero density only the“black hole” embedding exists. This is an embedding for which the probe brane isextended through the horizon of the bulk black hole (or the Poincare horizon if zero1We will be interested in the region of parameter space for which the resulting geometry is well-described by type IIA supergravity in ten dimensions.97temperature is considered). The instability of a compact, “Minkowski” embed-ding is intuitive: In the presence of finite density, and therefore finite electric flux,sources for the electric field are needed for a compact embedding. The analysis in[88] then shows that the available (bosonic) charged sources are strings connectingthe brane to the horizon, which inevitably pull the brane embedding towards thehorizon. One of the consequences of our construction is that this outcome maybe avoided in the presence of charged fermion sources. The reasoning behind thisis also intuitive: if the charged sources are fermions, the Pauli exclusion principledictates they form a Fermi surface. The resulting Fermi pressure counteracts thepull towards the horizon, potentially resulting in a stable configuration. Here wegive an example of such a configuration.The near horizon geometry of the D2-brane stack, at zero temperature, is:ds2 =−dt2+dx2+dy2u5/2+du2u3/2+u1/2(dS 23 + sin2(θ)dS˜ 23)(5.2.1)where we use the function 0< u< ∞ as the holographic radial coordinate with theboundary located at u = 0, and we set the spacetime radius of curvature to unity.The coordinates t,x,y parametrize the boundary field theory directions. In additionto the curved metric, the spacetime has a non-trivial dilaton profile and RR flux.The D6-brane is extended in the field theory and holographic radial directions,and wraps three of the six compact directions. It is convenient then to introducecoordinates such that the sphere wrapped by the D6-brane is S˜3. This sphere isfibered over the sphere S3 on which the brane is localized. The location of thebrane is specified by an angle2 θ , and the volume of S˜3 depends on that locationas indicated.The embedding of the D6-brane may then be specified by giving its location θas function of the holographic radial coordinate u. As explained in [50], for branesthat do not cross the horizon a more natural parametrization near the cap-off pointis provided by u(θ). Here we choose this parametrization globally, resulting in asomewhat unusual form of the DBI action. The coordinate θ ranges between itsvalue at the cap-off point θ = 0 and its asymptotic value θ = pi2 as u→ 0. In this2In the least action solution we expect the brane location in the other angular directions onS3 tostay constant.98parametrization the induced metric on the brane is:ds2induced =−dt2+dx2+dy2u5/2+du2u5/2(u2u′2+1)+ sin2(θ)√udS˜ 23 (5.2.2)From this one can determine [50] that the worldvolume caps off smoothly atfinite value of the radial coordinate if u(θ = 0) = u0 and u′(θ = 0) = 0, and thatthis point is reached when the radius of the S3 goes to zero, i.e. when θ = 0,as indicated. Near the boundary, u(pi2 ) = 0, the embedding function has the nearboundary expansion:u(θ)' m0(pi2−θ)+χ(pi2−θ)3+ ... (5.2.3)where m0 and χ are constants. For a constant worldvolume gauge field, a solutionfor the embedding equation with the desired properties is u(θ) = m0 cosθ .In addition to the embedding function, we will need to turn on the gauge po-tential on the D6-brane, which we denote by G(θ). The action for the two bosonicfields is then proportional to the DBI action:L =sin3(θ)√u′(θ)2+u(θ)2−u(θ)4G′(θ)2L3u(θ)5(5.2.4)where we have chosen α ′ = 12pi . Near the cap-off point smoothness requires thatG′(0) = 0, while near the boundary:G(θ)' µ+ρ(pi2−θ)2+ ... (5.2.5)Our boundary conditions near the asymptotic boundary are therefore G(pi2 ) = µand u′(pi2 ) = m0 where m0 and µ are parameters of our solution.5.2.2 Fermionic sectorWe now turn our attention to the fermionic sector. We work with fermions localizedin the field theory and radial directions, described by the Dirac action coupled tothe bosonic sector in a manner described below. In a “top-down” context, charged99fermions can arise when considering multiple probe D-BRANES. Consider for ex-ample an additional D6-brane with the same configuration as above, but localizedat θ = 0 . When the two D6-branes are separated we have a massless Abelian gaugefield (the “relative” gauge group) and charged fermions with respect to that gaugefield. In this setup, however, there are additional fields, including charged bosons.The presence of light charged bosons is likely to result in Bose-Einstein condensatebeing the dominant phase at low temperatures. However, this conclusion can beavoided by various additional couplings or other complications. As such couplingswill unnecessarily complicate our analysis, we take a phenomenological approach(commonly used in constructing gravity duals) and use the minimal matter contentand couplings required to construct the state we are interested in. We keep theabove “top-down” context only as a motivation for our construction.Our fermionic action is then as followsS =−iβ√−γ(ψ¯ΓMDMψ−m(θ)ψ¯ψ) (5.2.6)DM = ∂M +14ωabMΓab− iqAMΓM = ΓaeMam(θ) = mψ +u(θ)14 sin(θ)where M refers to bulk space-time indices and a,b to tangent space indices. Here qis the electric charge, γ is the determinant of the induced metric, and ω is the spinconnection. The coupling to the induced metric γab and gauge field follows fromsymmetries, which determine the form of the covariant derivative we use, DM. Inaddition to the bare Dirac mass mψ , we choose to add a Yukawa coupling givingthe fermions an effective mass proportional to the radius of the S˜3. This couplingis motivated by the “top-down” context described above. The tuneable parameterβ controls the backreaction of the fermions on the bosonic sector3, and is typicallytake to be fairly small. Note that in the brane setup β ∝ gstr since it arises as theratio of the coefficient of the Dirac action to the brane tension (which we chose tonormalize to unity in the bosonic action Equation 5.2.4.3The parameter β is defined only with respect to a specific normalization of the fermionic fields,which we choose as∫ pi/20 dθψ¯ψ = 1.100To solve the fermionic equations we follow the procedure first outlined in [81,92], but using the notations of [47, 80, 119]. First, it is convenient to rescale thefermions in order to remove the spin connection from the Dirac equation:ψ = (−γγuu)− 14 e−iωt+ikixiΦ (5.2.7)where γuu refers to the radial component of the induced metric given in equationEquation 5.2.2 and w,ki are the frequency and spatial momenta, respectively (withi = x,y). We then divide the four components of the Dirac spinor to normalizableand non-normalizable modes. Specifically, if we define4 P± = 12(1±Γu) thenφ± ≡ P±Φ≡(y±z±)(5.2.8)are normalizable and non-normalizable modes of the Dirac fermion. Furthermore,we can use rotational invariance in the boundary directions to choose the momen-tum kx = k,ky = 0. This choice allows us to organize the four Dirac equationsas a decoupled pair of two ordinary differential equations. Choosing real gammamatrices as in [47, 80, 119], the decoupled components can be organized intoΦ1 =(iy−z+), Φ2 =(−iz−y+)(5.2.9)It is therefore sufficient to consider only one of Φ1 and Φ2 when construct-ing our state and calculating correlation functions. Information regarding the otherspinor may be extracted using rotational invariance as in [47, 80]. We thereforechoose to work with, in a slight change of notation, f¯ = ( f1, f2)≡ (z−,y+), reduc-ing the fermionic problem to the solution of two ordinary differential equations.We now consider the boundary conditions for the fermionic fields ( f1, f2). Nearthe asymptotic boundary there are two independent modes, normalizable and non-normalizable. For the former, the component f1 is constant near the boundary atθ = pi2 , and f2 is then determined by the equations of motion. For the latter, non-normalizable mode, the roles of f1 and f2 are interchanged. As we are seeking4Underlined indices for the gamma matrices are tangent space indices.101normalizable solutions we require f2(pi2 ) = 0.Turning to the behaviour of the fermions in the IR: Conservation of the sym-plectic norm requires that either f1 or f2 must be zero at the cap-off. This statementcan be verified by integrating the symplectic flux ψ†Γ0Γµψnµ over the hypersur-face located at u = u0. To obtain a non-trivial solution we then set f1(0) = 0 forregularity in the interior. Since we have two first-order ordinary differential equa-tions, our boundary value problem is then well-posed.5.2.3 Parameters and limitsOur equations and boundary conditions have the following parameters:• The chemical potential, µ .• The source term for the scalar field dual to the embedding function, m0.• The fermion bare mass mψ .• The electric charge q.• The relative strength of the contribution of the Dirac and DBI actions to thetotal action. This is determined by the parameter β .In our numerical construction, described below, we vary all five parametersindependently. We divide these into two groups- the thermodynamic variables,(µ,m0), and the mass and coupling terms which define our theory, (β ,m0,mψ).The physical significance of these parameters is as follows:• Varying the chemical potential controls the fermionic density on the brane.As we are working at zero temperature the chemical potential sets the bulkFermi energy.• Varying m0 controls how far the embedding proceeds into the bulk beforecapping off. Increasing m0 moves the the cap-off point further into the IR.As such embeddings require larger charge densities to support them we willsee that increasing m0 also has the effect of increasing the charge density.102• The parameter β controls the backreaction of the fermionic charge densityon the bosonic fields, i.e. β = 0 corresponds to decoupled fermions5. Herewe work with a finite but small β .• It is convenient to rescale the matter fields and charge as H¯ →H ε , q→ eε ,whereH represents the gauge field or fermion fields and e is fixed. We thentrade the charge q for the parameter ε , small ε corresponds to large charge.The limit ε → 0 is the commonly used probe limit, as used for example inconstructing holographic superconductors [59]. Setting ε = 0 decouples theembedding function from the Maxwell-Dirac sector. Here we work withsmall but finite ε .• Varying the bare mass terms of the fermions mψ corresponds to setting theirmass at the cap-off point.We comment further below on the parameter dependence of the state we con-struct, and the limits and range of values of these parameters in our numericalsimulations.5.3 Bulk Fermi Surface5.3.1 Iteration procedureThe main novelty in solving the equations of our setup is the implementation ofFermi statistics. The requirement that the fermions form a Fermi surface is a non-local constraint, effectively rendering the problem an integro-differential system ofequations, instead of a set of local ordinary differential equations. We thereforesolve our system, as in [119], by an iterative process we now describe. Furtherdetails of our implementation are found in Section C.2.For every step in the iteration process, the fermion equations are formulated ina background of the bosonic fields. One solves the Dirac equation in that back-ground to find the complete set of eigenstates of the Dirac operator. The state ofthe fermions in a fixed chemical potential (and zero temperature) is a filled Fermi5This is similar to [47] where the Dirac equation was considered on a RN black hole background.103surface: all states with energy below µ are filled. These eigenstates come in bands;we work with parameter ranges such that only a small number of these bands (ap-proximately 1 to 10) are filled. This has the interpretation of multiple (but orderone in the large N limit) Fermi surfaces in the dual QFT. The inclusion of severalbands is necessary when one tunes parameters towards states of larger charge den-sities. The state of the fermions is iteratively adjusted mainly through the changein the eigenstates of the Dirac operator.Given the state of the fermions, the bosonic equations are sourced by specificfermion bilinears, obtained from varying the action Equation 5.2.6 with respect tothe bosonic fields (the specific form of these bilinears and further details regardingtheir properties can be found in Section C.1). Briefly, we identify these sources asthe charge density, Q, which sources the gauge equation and embedding equation.In addition the embedding equation is sourced via effective stress energy terms Tr,and TM which we label as the “radial” and “Minkowski” stresses. As in [119], weevaluate these sources at leading order in 1N , i.e. in the classical limit in the bulk.This approximation does not include the higher order (and much more complex)renormalization of the coupling constants in the brane action. We comment belowon regimes of solutions where such higher order effects may become relevant andtheir potential implications for the iterative solution.Once the adjusted bosonic background is obtained, we turn back to the fermioneigenvalue problem, and iterate to convergence. Details of our numerical algo-rithm can be found in Section C.2: both the fermionic eigenvalue problem andthe bosonic boundary value problem were discretized using pseudo-spectral col-location methods on a Chebyshev grid. The iterative procedure described abovewas repeated until the residuals for the bosonic system were driven to sufficientlylow values and the change in solution between successive steps, for the gauge andembedding functions, was sufficiently small.This solution method was found to converge for regions of parameter spacewhere the backreaction was small relative to the radial scale of the probe limit em-bedding. We will therefore restrict our investigations to regimes of small densities,leaving investigation of higher density regimes for future work.1045.3.2 Solutions in the probe limitIn this section we examine the behaviour of the system in the probe limit, corre-sponding to taking ε = 0. This decouples the gauge and embedding equations andfreezes the embedding function to be u(θ) = m0 cos(θ). Allowing for the probelimit, our model still has four tuneable parameters- namely β , m0, mψ and µ . Tostudy the effect of these parameters, we vary individual parameters while leavingthe others fixed. In order to visualize the bulk solutions we display the gauge field,the fermionic dispersion relationship, and the charge Q, for each of the parametervariations.We start by varying the chemical potential µ . Plots for bulk quantities for var-ious values of µ are displayed inFigure 5.1. We note that, as expected, increasingµ results in larger charge densities. Note that the charge density is finite at thecap-off (unlike the case with bosonic charge carriers, where the cap-off point sup-ports all of the charge). This is an indication that such a charge density may beable support the compact embedding once we include backreaction. As expectedthe number of bands increases with µ , for large µ up to 10 bands may be filled. Asthese higher bands correspond to higher modes of the bulk fermion eigenfunctionsthey contribute a modulated component to the charge density. This may be seen inthe above figures.We now examine the effects of moving within our parameter space of theories,defined by (mψ ,m0,β ). The results for variations of the bare mass term, mψ , areshown in Figure 5.2. We see that decreasing the bare mass has the effect of pushingus towards a regime of higher fermion density. A reduction of mψ from an initialhigh value results in a rapid increase in the peak amplitude of the bulk chargedensity. However for sufficiently low values of mψ the charge density profile isseen to broaden and shift towards the UV geometry. This results in an enhancementof the charge density of the dual QFT at low mψ .We can look at the effects of adjusting our (frozen) embedding cap off pointby changing m0. This simulates the expected adjustment in the position of theembedding cap-off in response to the finite charge density, once backreaction isincluded. From Figure 5.3 we see that larger values of m0 correspond to largerfermion densities, and electric field strengths.1050 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.050Gauge field, GθG  mu=−12.4839mu=−16.0612mu=−19.4194mu=−23.3617mu=−26.35490 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6050100150200250300350400450Charge densityθQ11 + Q220 1 2 3 4 5 6−1.4−1.2−1−0.8−0.6−0.4−0.200.2Dispersion relationship low |µ|kω  µ=−13.14090 5 10 15 20 25 30−18−16−14−12−10−8−6−4−202Dispersion relationship high |µ|kω  µ=−29.1291Figure 5.1: Bulk field profiles resulting from varying µ with m0 = 1, β =−0.01 and mψ = 10. We note that the shape of the gauge field profiledoes not change greatly relative to the scale set by the chemical poten-tial, as µ is increased. This is illustrated by subtracting the value of thechemical potential in each case. The dispersion relationship curves arefor filled states lying below the Fermi surface.1060 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−15.73−15.728−15.726−15.724−15.722−15.72−15.718−15.716−15.714−15.712−15.71Gauge field, GθG  mψ=15mψ=13.5mψ=12mψ=10.5mψ=9mψ=7.5mψ=6mψ=4.5mψ=3mψ=1.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020406080100120140Charge densityθQ11 + Q220 0.5 1 1.5 2 2.5 3 3.5 4−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.05Dispersion relationship high mψkω  mψ=13.50 2 4 6 8 10 12 14 16−14−12−10−8−6−4−202Dispersion relationship low mψkω  mψ=1Figure 5.2: Bulk field profiles resulting from varying mψ with m0 = 1, µ =−15.7154, and β = −0.001. We see that increasing the bare mass hasthe effect of reducing the number of filled states, the amplitude of thegauge field and the contributions to the bulk charge. We also note that asmψ decreases the ratio of width to amplitude of the bulk charge densityincreases. This results in amplification of the charge densities in thedual QFT.1070 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−15.95−15.9−15.85−15.8−15.75−15.7Gauge field, GθG  m0=1.9m0=1.6m0=1.4m0=1.2m0=1m0=0.80 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6050100150200250300Charge densityθQ11 + Q220 0.2 0.4 0.6 0.8 1 1.2 1.4−0.05−0.04−0.03−0.02−0.0100.01Dispersion relationship low m0kω  m0=0.80 2 4 6 8 10 12 14 16−12−10−8−6−4−202Dispersion relationship high m0kω  m0=1.9Figure 5.3: The bulk field profiles resulting from varying m0 with β =−0.001, µ = −15.7154 and mψ = 10. We note that increasing m0 hasthe opposite effect to increasing mφ as it pushes the solution to regimesof higher fermion density. It can be seen the behaviour of the bulk gaugefield and charge density under the variation of µ and m0 is qualitativelysimilar. This is reflected in the behaviour of QFT quantities.1080 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−16.15−16.1−16.05−16−15.95−15.9−15.85−15.8−15.75−15.7Gauge field, GθG  β=−1.5849e−05β=−0.0001β=−0.00063096β=−0.0039811β=−0.025119β=−0.158490 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60102030405060Charge densityθQ11 + Q220 2 4 6 8 10 12−4−3.5−3−2.5−2−1.5−1−0.500.5Dispersion relationship high |β|kω  β=−0.398110 2 4 6 8 10 12−5−4−3−2−101Dispersion relationship low |β|kω  β=−2.5119e−06Figure 5.4: Varying β with m0 = 1, mψ = 10, µ =−15.7154. We see that in-creasing the magnitude of β has the effect of the increasing the bulkcharge density and correspondingly the strength of the bulk electricfield.Finally, turning our attention to β we examine the influence of the strength ofthe fermion-boson coupling in Figure 5.4. Increasing the magnitude of β produceslarger bulk charge densities and electric fields.5.3.3 Including backreactionWe now consider the effects of backreaction in our system via tuning ε to non-zero values. We now have a five parameter family of solutions characterized by(µ,m0,mψ ,β ,ε). In analogy to the previous section we investigate the influence1090 0.2 0.4 0.6 0.8 1 1.2 1.4 1.600.511.522.53x 10−5Change in gauge field, G,from probe limitθG  mu=−12.4839mu=−16.0612mu=−18.9814mu=−21.9016mu=−24.8218mu=−27.742s0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−14−12−10−8−6−4−202x 10−6Change in embedding function, u,from probe limitθu0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.025−0.02−0.015−0.01−0.00500.005Change in charge densityfrom probe limitθQ11 + Q220 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−300−250−200−150−100−50050TrθQ21 + Q120 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−10000−8000−6000−4000−200002000TMθL11 + L22 + P11 − P220 5 10 15 20 25 30 352345678910x 10−4 Change in dispersion high µkω  µ=−31.9033Figure 5.5: Varying µ with backreaction included, and normalizing our solu-tions with respect to the probe limit. The other parameters were set asβ =−0.01, ε = 0.1, m0 = 1, mψ = 10. Electric field strength and chargedensity are shifted further away from the embedding cap-off relative tothe probe limit. 110of each individual parameter on the bulk solutions when the others are kept fixed.In order to determine the significance of backreaction we normalize our solutionswhere appropriate by subtracting off the probe limit background.We are particularly interested in examining whether the embedding tends toevolve, when changing parameters, in a direction where the volume in the IR in-creases significantly. If, in a limiting case, the IR geometry were to tend towardsbecoming non-compact, it may signify the breakdown of bulk perturbation theory.In the dual QFT this phenomena would signal the breakdown of the largeN expan-sion. We will illustrate in Section 5.4.2 below that such a breakdown is necessaryfor large quasiparticle scattering rates.As in the previous section, we first consider changes in our thermodynamicvariables. Our ensemble is now defined via (µ,m0), as encoded in the boundaryconditions for the relevant bulk fields. In Figure 5.5 we plot the change in thebulk fields displayed in Figure 5.1 once backreaction is included. We note that thecharge density decreases slightly in the vicinity of the cap-off once backreaction isincluded. This has the effect of decreasing the electric field strength in the cap-offregion and allowing it to retract slightly towards the UV. This change is sourced byTr and TM which we also display below. These effects become more pronouncedas |µ| increases. The net result is that the transverse sphere collapses to a pointslightly sooner then in the probe limit case, but still does so at a finite value as aresult of support provided by the Fermi pressure. We also note that the energy ofthe filled states is decreased relative to the probe limit. A similar story emergesas we follow the branch of solutions parameterized by m0, as seen in Figure 5.6.Electric field and charge density profiles are shifted slightly towards the mid-regionof the geometry while the cap off itself retracts towards the conformal boundary.Turning to mψ we see that, for sufficiently large values of this parameter,changes to the bulk fields due to backreaction are suppressed. Initially, decreasingmψ has a similar effect on the sources for the embedding function, charge densityand electric fields as increasing m0 or µ . However at a critical value (for the choiceof parameters in Figure 5.7 this occurs at mψ ' 6) a qualitative change in behaviouroccurs. The rate of change of the peak modulation relative to the probe limit slowsdown and saturates. Past this point the effects of backreaction are seen to extendfurther and further towards the UV boundary while their magnitude decreases.1110 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.0100.010.020.030.040.05Change in gauge field, G,from probe limitθG  m0=1.9m0=1.7m0=1.6m0=1.5m0=1.4m0=1.2m0=1.1m0=1m0=0.9m0=0.8m0=0.70 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.14−0.12−0.1−0.08−0.06−0.04−0.0200.02Change in embedding function, u,from probe limitθu0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−45−40−35−30−25−20−15−10−505Change in charge densityfrom probe limitθQ11 + Q220 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−140−120−100−80−60−40−20020TrθQ21 + Q120 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−3000−2500−2000−1500−1000−5000500TMθL11 + L22 + P11 − P220 2 4 6 8 10 12 140.20.30.40.50.60.70.80.911.11.2Change in dispersion relationship high m0kω  m0=1.9Figure 5.6: Varying m0 with backreaction included, and normalizing oursolutions with respect to the probe limit. The couplings are set asµ = −15.7154,ε = 0.1,β = −0.001,mψ = 10. We note the qualita-tive similarity to the changes observed in the profiles at fixed µ as seenin Figure 5.5.1120 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1012345678x 10−5Change in gauge field, G,from probe limitθG  mψ=15mψ=13.5mψ=12mψ=10.5mψ=9mψ=7.5mψ=6mψ=4.5mψ=3mψ=1.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−20−15−10−505x 10−4Change in embedding function, u,from probe limitθu0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.45−0.4−0.35−0.3−0.25−0.2−0.15−0.1−0.0500.05Change in charge densityfrom probe limitθQ11 + Q220 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−60−50−40−30−20−10010TrθQ21 + Q120 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1800−1600−1400−1200−1000−800−600−400−2000200TMθL11 + L22 + P11 − P220 2 4 6 8 10 12 14 1600.0050.010.0150.020.0250.030.035Change in dispersion low mψkω  mψ=1Figure 5.7: Varying mψ with µ = −15.7154, m0 = 1, β = −0.001, ε = 0.1.Here we plot the change in the embedding function, gauge field andcharge density relative to the probe limit. We also plot the the un-subtracted components of the sources for the embedding function, Trand TM. 113Next, in Figure 5.8 and Figure 5.9 we examine the effects of backreaction forvariations of β and ε, respectively. As these control the coupling of the matter sec-tor fields to the embedding function, increasing the magnitude of either parameterserves to accentuate the effects of backreaction. The qualitative nature of this be-haviour is similar to that that observed for the other parameter variations describedpreviously. In the case of Figure 5.8 it is interesting to note the relative lack ofmodulation of the profiles of the embedding sources or the charge density, in com-parison to variations involving m0, µ or mψ . This may be attributed to the relativelyminor changes which occur in the spectrum of filled states. The energy of the filledbands changes as β is varied however their number and qualitative shape does not.A similar pattern may be observed for ε variations in Figure 5.9.From the above analysis we conclude that the incorporation of backreaction inthe model has two principle consequences:• Due to the finite charge density at the cap-off and the associated Fermi pres-sure Minkowski embedding are now possible. The effect of backreaction ismodify the geometry in the region near where the geometry caps off.• While backreaction remains small its effect on the bulk fields is to shift thecharge density distribution and associated electric field slightly in the direc-tion of the conformal boundary. The fermions are less tightly bound then inthe probe limit.Finally, in Figure 5.10 we examine the change in the volume element from itsthe probe limit value, for the parameter variations described previously. We seethat generically the inclusion of backreaction leads to an increase in the volumeelement near the cap-off. This difference increases for increasing |µ|, m0, |β |and |ε| corresponding to states of larger charge density. For variations of mψ thechange of the volume element is maximized when the change in the embeddingitself is greatest (low mψ ). These results provide a tentative indication that the highdensity limit may be an interesting regime to probe for possible non-Fermi liquidbehaviour.1140 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.500.511.522.5Change in gauge field, G,from probe limitθG  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.12−0.1−0.08−0.06−0.04−0.0200.02Change in embedding function, u,from probe limitθu  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−90−80−70−60−50−40−30−20−10010Change in charge densityfrom probe limitθQ11 + Q22  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−40−35−30−25−20−15−10−505TrθQ21 + Q12  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−300−250−200−150−100−50050TMθL11 + L22 + P11 − P22  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 2 4 6 8 10 1266.577.588.59x 10−6Change in dispersion relationship low βkω  β=−1.5849e−06Figure 5.8: Varying β , with µ =−15.7154, m0 = 1, mψ = 10, ε = 0.1.1150 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−1012345678x 10−4Change in gauge field, G,from probe limitθG  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.000177830 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−0.035−0.03−0.025−0.02−0.015−0.01−0.00500.005Change in embedding function, u,from probe limitθu  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.000177830 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−8−7−6−5−4−3−2−101Change in charge densityfrom probe limitθQ11 + Q22  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.000177830 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−40−35−30−25−20−15−10−505TrθQ21 + Q12  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.000177830 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−300−250−200−150−100−50050TMθL11 + L22 + P11 − P22  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.000177830 1 2 3 4 5 6 7 8 9 100.320.340.360.380.40.420.44Change in dispersion relationship high εkω  ε=1Figure 5.9: Varying ε with β =−0.001, m0 = 1, mψ = 10 and µ =−15.7154.Increasing the value of ε has a similar effect to increasing the magnitudeof m0 or µ, resulting in lower charge densities near the cap off and aslight movement of this cap-off towards the IR.1165.4 Boundary FermionsWith the above constructed bulk solutions, we can proceed to probe the behaviourof the dual QFT in the state we constructed. Our principle aim in this section isto demonstrate that the states we are considering are those of a Fermi liquid. Wealso wish to keep in mind future generalizations which could produce non-Fermiliquid behaviour. To this end we study the equation of state as a function of variousparameters and examine the analytic properties of the fermion Green’s function.5.4.1 Fermion densityIn Figure 5.11 we examine the fermion density of the boundary theory as a functionof the various parameters discussed previously. We start by discussing the equationof state: the dependence of the density on the chemical potential. Plotting |ρ|versus |µ| we see that the system is gapped, with the mass of boundary fermionsdetermined by the intercept of that plot. For the parameters chosen in Figure 5.11mQFT ' 12.2173.One of the basic signals of a Fermi surface is that the state is compressible, i.ethat dρdµ 6= 0 for all µ for which the Fermi surface exists (i.e. for chemical potentialabove the gap). Our plot demonstrate that our model indeed has this feature. Wealso fit the asymptotic behaviour of the curve to the form |ρ| ∝ b|µ|a , with b =5.945e−5 and a= 3.064 for the choice of parameters plotted. Note that for a singlespecies of fermions in 2+1 dimensions, the exponent a ranges between one (non-relativistic fermion) to two (relativistic fermion), in the regime of asymptoticallylarge |µ|. The asymptotic scaling we find is a result of the multiple species offermions which exist in our model due to the presence of multiple bulk bands. Atasymptotically large chemical potential large number of these bands are occupied.Examining the density as a function of other parameters is also interesting. Wefind that the density decreases with mψ and increases with m0. This is in line withour expectations from the observed behaviour of the bulk fields in Figure 5.2 andFigure 5.3. Zero density was found to occur at a finite value of m0 which we labelas mcritical0 (in the case plotted in Figure 5.11 mcritical0 ' 0.701). This indicates thatcompact embedding solutions only exist for solutions which penetrate sufficientlyfar into the bulk. Fitting the density to the form d(m0−mcritical0 )c we find c =1170.05791, d = 3.77. Similarly mcriticalψ was found to be ' 13.9981 in our currentcase.The last two graphs indicate that |ρ| increases with |β | and |ε| in some non-trivial manner. Again this is to be expected from the plots of the bulk fields inFigure 5.4 and Figure 5.9.5.4.2 Retarded Green’s functionThe retarded Green’s function in the vicinity of the Fermi surface has the formGR(ω,k) =Zω− vF(k− kF)+Σ(ω,k) (5.4.1)The quasiparticle decay rate is determined by the self-energy Σ. In the case of aLandau-Fermi liquid it is known to universally scale as Σ= iΓ2 ∼ iw2, and thereforethe quasiparticle lifetime diverges as the Fermi surface is approached. In the caseof non-Fermi liquids Σ scales faster than ω6 at low frequencies. This indicates thata quasiparticles are not a good description of the physics close to the Fermi surface.The first step is to identity the presence of a Fermi surface in the dual QFT. Aswe noted previously there are two distinct fermion modes as u→ 0:1. limθ→ pi2 ( f1, f2) = (c0,0)2. limθ→ pi2 ( f1, f2) = (0,d0).A priori both quantizations of the fermions are possible, however in constructingour background we picked the first by setting f2(pi/2) = 0. As our backgroundshould be constructed from purely normalizable modes we label mode (1) as thenormalizable mode. Mode (2) should then be considered as the source which weturn on to perturb the system.As the retarded Green’s function is proportional to the ratio of response tosource, we wish to calculate the ratio of the asymptotic values of the linearized per-turbations δ f1 and δ f2. For the purpose of identifying the location of the boundaryFermi surface it is sufficient to consider only the linearized fermionic equations.6There will be an additional logarithmic term in the case of a Marginal Fermi Liquid.118In order to identify the singularity in the Green’s function it is convenient to useshooting techniques. Once the singularity in the Green’s function was identified,the behaviour in the vicinity of the pole in the k,w plane can be seen inFigure 5.12.We note immediately that our Green’s function is purely real and therefore lacksany information about the quasiparticle decay rate. Mechanically this is the resultof the fact that our Dirac equation, Γ matrix algebra and eigenfunctions are purelyreal. This, together with the fact that our boundary conditions in IR are simpleregularity conditions, ensures that our Green’s function is real7. Less mechanically,the fact that we lack a black hole horizon means we do not have a mechanism fordissipation at leading order in the 1N expansion. In order to calculate the decay rateit would be necessary to calculate 1-loop diagram (for references with such 1-loopcalculations, see [28, 29, 46]). The fact that the decay rate will be parametricallysmall indicates that this phase is a Fermi-like liquid, as long as perturbation theoryis valid.Indeed, one of the motivations for the present work is identifying potentialsources for such a breakdown, for example the geometry becoming non-compactin the IR. This intuition ties in with [47] where the infinite proper distance of theAdS2×R2 throat geometry is dual to the low energy bosonic modes necessary toprovide fermionic quasi-particle dissipation. We hope to return to this direction inthe future.5.5 Concluding RemarksIn this study we have constructed a phase of holographic matter with Fermi liquidlike behaviour, by solving for finite density fermion configurations on Minkowskiembedding of a probe D-brane.Possible generalizations of this setup may be useful in the study of non-Fermiliquids. We hope that our study will help identify the regimes of parameter spacewhere such behaviour may be expected. For example, a sufficiently large defor-mation of the embedding may introduce large renormalization effects, invalidatingthe purely classical approximation used here. In such regime a more careful study,7In other works such as [46, 47] the ingoing boundary conditions associated with a black holehorizon render the solution complex.119along the lines of [5, 6], would be needed to determine the nature of the fermionicstate on the D-brane.More generality, the ability to construct a finite density Minkowski brane con-figurations could be useful in a range of holographic models, as those are expectedto be qualitatively different than the finite density black hole embeddings. Forexample, the compact nature of the world volume geometry means the charged de-grees of freedom lack an efficient mechanism of dissipation, therefore such phasescan be useful in exploring the physics of holographic insulators.1200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.511.522.533.54x 10−5Change in volume elementrelative to probe limitθVolume element  mu=−12.4839mu=−15.2581mu=−18.0323mu=−20.8065mu=−23.5807mu=−27.7420 0.2 0.4 0.6 0.8 1 1.2 1.4−0.0500.050.10.150.2Change in volume elementrelative to probe limitθVolume element  m0=1.9m0=1.7m0=1.6m0=1.5m0=1.4m0=1.2m0=1.1m0=1m0=0.9m0=0.8m0=0.70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−101234567x 10−3Change in volume elementrelative to probe limitθVolume element  mψ=15mψ=13.5mψ=12mψ=10.5mψ=9mψ=7.5mψ=6mψ=4.5mψ=3mψ=1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.020.040.060.080.10.120.140.160.180.2Change in volume elementrelative to probe limitθVolume element  β=−1.5849e−06β=−1e−05β=−6.3096e−05β=−0.00039811β=−0.0025119β=−0.015849β=−0.1β=−0.630960 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.010.020.030.040.050.06Change in volume elementrelative to probe limitθVolume element  epsilon=1epsilon=0.17783epsilon=0.031623epsilon=0.0056234epsilon=0.001epsilon=0.00017783Figure 5.10: The change in volume near the embedding cap off. Higherfermion densities tend to increase the volume element, relative to theprobe limit.12110 15 20 25 30 3500.10.20.30.40.50.60.70.80.9Density as a function ofchemical potentialµ−ρ  Density vs. µ0.8 1 1.2 1.4 1.6 1.8 2−0.0200.020.040.060.080.10.12Density as a function of m0m0−ρ  Density vs. m00 5 10 15−505101520x 10−3Density as a function ofmψ−couplingmψ−ρ  Density vs. mψ−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−0.0500.050.10.150.20.250.3−β−ρDensity as a function of β  Density vs. β0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.64.74.84.955.15.25.35.45.5x 10−4ε−ρDensity as a function of ε  Density vs. εFigure 5.11: The magnitude of the boundary charge density for the variousparameter ranges discussed in the text. The plots are calculated inthe backreacted case, though they are not greatly influenced by thebackreaction.12200.511.522.5x 10−500.511.522.5x 10−5−0.100.10.20.30.40.50.6k−kfSingularity inretarded Green  functionwGreen functionFigure 5.12: The multiple filled bands for bulk fermions translate into mul-tiple Fermi surfaces in the boundary theory. Each of these Fermi sur-faces is associated with a pole in the retarded Green’s function. Herewe plot one illustrative example of the structure of the Green’s functionin the (k,w) plane in the region near one Fermi surface at k f = 15.3676.123Chapter 6Spatial Modulation andConductivities in EffectiveHolographic Theories6.1 Introduction and OutlineIn this chapter we resume our exploration of non-translationally invariant phasesof condensed matter with the goal of illuminating aspects of metallic, incoherent-metallic, and insulating phases within the quantum critical regime. To examinethe properties of, and transitions between, these phases we will need to probe thetransport properties of the system. As such, consider the linear response functionswhich may be efficiently computed in the holographic context. In the absence ofdissipation the low frequency behaviour of correlation functions often display IRdivergences. While some of these can be cured by examining the theory at finitetemperature (effectively using a thermal IR regulator), there are some issues thatare incurable by this simple approach.A simple case in point is the physics of electrical conductivity. Naively, wemight try to compute this quantity in a translationally invariant system by monitor-ing the response to turning on an external, time dependent electric field. Using theKronig-Kramers relations one would then find that momentum conservation results124in a divergent zero frequency value of the imaginary part of the conductivity. Asa consequence recent efforts have been focused on constructing models with morerealistic conductivity behaviour. This in turn is motivated by the need to betterunderstand the experimental observation of metal/incoherent-metal/insulator tran-sitions in the phase diagram of high temperature superconductor materials [12].1The first models for realistic behaviour of (thermo)-electrical conductivitiesin holography was accomplished in [72, 73]. These authors considered situa-tions where the spatial homogeneity is explicitly broken by a background latticeof sources, and studied thermo-electric transport in the resulting holographic dualbackground. By use of the lattice, they were able to demonstrate that the low fre-quency conductivity was finite, and well fitted in their set-up by the Drude form.More curiously, their analysis revealed a mid-range scaling of the AC conductiv-ity with frequency, which furthermore agreed with experimental results in cupratesystems [127].Since this seminal work, various groups have attempted to understand thethermo-electric properties of holographic systems. One of the key aspects to un-derstanding this phenomenon is to ascertain the potential low energy behaviour ofstrongly coupled systems subjected to inhomogeneous sources for relevant opera-tors. A classification of emergent IR phases crucially provides one with a pictureof what to expect for the transport properties of the system at low energies. Ex-tensive work has already been carried out in characterizing the possible IR geome-tries which may emerge when translational invariance is relaxed; see for example[26, 43] where the IR flow of perturbative inhomogeneous modes was analyzed.We now have a reasonable understanding for the spectrum of possibilities that canoccur in holographic systems, with low energy behaviour ranging from metallic(with various dressing) to insulating. Part of the motivation for the present workwas to get a better picture of the landscape of possibilities within the framework ofbottom-up holography.We should also note other innovative approaches towards modelling aspects ofinhomogeneous systems. These include the techniques of massive gravity [7, 15,16, 25, 128] and Q-lattices [8, 37–39, 41, 52] (cf., also the memory function ap-1 A nice summary of the issues involved in characterizing such behaviour in many-body systemscan be found in [55].125proach [94, 95]). The Q-lattices are particularly appealing as, though translationalinvariance is broken, the metric remains homogeneous and the equations of mo-tion remain relatively tractable ODEs. Unfortunately the fine tuned nature of thissymmetry raises the possibility that some of the behaviour observed in the result-ing solutions may be equally non-generic. Likewise while much progress has beenmade using the massive gravity approach, questions remain about the interpretationof such models in the dual field theory.With these issues in mind let us turn back to explicit breaking of translationalinvariance in one of the spatial directions of the dual field theory, as described in[72, 73] (see also [91]). As mentioned above, the introduction of the holographiclattice and solutions of the resulting PDEs mitigates the zero frequency delta func-tion in the conductivity, leading to a conventional Drude form for low frequencies.An extremely thorough examination of the subject in [40] combines technologygained from Q-lattice calculations with the numerical approach of [72, 73, 91] toexamine the electric, thermoelectric and thermal conductivity. Among the resultsof this analysis were the confirmation of the Drude regime for the AC conductivityand the existence intermediate resonance peaks. It was also noted that in the modelunder consideration no evidence of a mid-IR scaling regime was discovered.In this note we aim to build on the work of [40, 73, 91] by exploring a par-ticular class of Einstein-Maxwell-Dilaton models. These models are characterizedby two functions of the scalar dilaton field Φ – these parameterize the scalar self-interaction through a scalar potential and a gauge coupling function, V (Φ) andZ(Φ) respectively. This parameterization allows us to access a range of effectivefield theories which are of interest in various condensed matter applications. Forexample, examination of the homogeneous solutions has shown that these modelsare good candidates for metal/insulator transitions upon breaking of translationalinvariance [20]. The range of allowed IR phases can be understood in terms of aneffective near horizon potential, obtained by approximating V (Φ). One finds thatthis potential modulates the spectrum of ingoing excitations near the horizon fromcontinuous (metallic) to discrete (insulating) as a function of parameters. Qual-itatively, the physical features of these models can be understood by noting thatscalar potential interpolates between zero at the conformal boundary and a run-away behaviour in the near-horizon region. Similarly the gauge coupling starts at126some finite value near the conformal boundary and undergoes exponential growthor decay towards the horizon.We will examine the electric conductivity of these models as a function of pa-rameters. We employ two lines of attack, both of which involve first numericallyfinding the bulk solution to sourced inhomogeneities in the UV. This holographiclattice solution can then be explored for transport. The simplest analysis is then toextract the DC conductivities via a membrane paradigme-esque formula using thetechniques of [40] (cf., Section 6.3). This approach has the advantage that we onlyneed knowledge of the bulk solution and therefore gives us a simple way to probethe IR phase by analyzing the DC conductivity [73]. Computing the AC conductiv-ity however requires that we also have the solution of the linearized perturbationequations around the numerically constructed background [73]. We carry out thisexercise to obtain the full frequency dependent transport, in the process using theaforementioned DC conductivity as a non-trivial check on the results we obtain. Wediscover the existence of metal-insulator transitions as a function of parameters andalso, in finely tuned cases, the potential presence of a mid-IR scaling regime. Wethen examine the persistence of these behaviours as a function of parameters. Wefind that that the phase changes we discover via monitoring the behaviour of theresponse functions correspond to inhomogeneity mediated changes in the form ofthe scalar potential in the IR.The outline of the paper is as follows. In Section 6.2 we present a basic reviewof the holographic set-up, pausing to note the ingredients we pick in our modeland their potential effects on the low energy dynamics of the system. We alsogive a short synopsis of the numerical scheme we employ to study the physicaltransport. In Section 6.3 we revisit the arguments of [40] to directly extract thezero frequency conductivities in terms of horizon data. In course of this analysiswe take the opportunity to explain the relation between the various conductivities inthe hydrodynamic limit (which corresponds to ω T and a suitably dilute lattice).We illustrate that in this regime there is a single transport coefficient which canbe taken to be the electrical conductivity. In Section 6.4 we present our resultsfor the various transport coefficients. We also comment on the various phasesand scaling regimes which we observe as we explore a representative subspace ofconfigurations for the scalar potential and Maxwell coupling. We conclude with127some discussion in Section 6.5. Some technical points about our numerics arecollated in Section D.1.6.2 SetupWe start by outlining the holographic set-up we use to explore the physics of con-ductivity in 2+1 dimensional quantum critical systems. We explain the basic in-gredients we employ in our phenomenological modelling as well as the salientfeatures of the control functions we introduce in parameterizing the holographicsystem. Following this discussion we go on to describe the numerical scheme usedto construct the gravitational solutions of interest and the computation of the phys-ical conductivities therein.6.2.1 BackgroundWe take a bottom up, phenomenological approach to holography, and following[40, 73, 91] we choose a model with the minimal ingredients necessary to calcu-late conductivities in a 2+1 dimensional quantum critical theory. We assume thatthe field theory is holographically dual to gravitational dynamics in an asymptoti-cally ADS spacetime. The model for the holographic dynamics is simply Einstein-Maxwell theory, which we couple to an additional neutral scalar field, with poten-tial term, V (Φ), and a gauge coupling function, Z(Φ). The scalar field allows usto discuss models with more general behaviour in the IR than the local criticalitycharacterized by the RN black hole. This results in an action of the form (setting`AdS = 1):S =∫d4x√−g(R+6− 14Z(Φ)Fab Fab−∇aΦ∇aΦ− 12 V (Φ)). (6.2.1)The dynamical equations of motion which we will solve are then simply∇a (Z(Φ)Fab ) = 0 , (6.2.2)∇a∇aΦ−V ′(Φ) = 0 ,Rab+3gab−(∇aΦ∇bΦ+12[12gabV (Φ)+Z(Φ)(FacF cb −14gabFcdFcd)])= 0 .128The field theory will be taken to live in Minkowski spacetime and we shall thuswork with the conventional parameterization of ADS geometry in Poincare´-like co-ordinates. The radial coordinate is taken to be z ∈R+ and the conformal boundarylocated at z = 0.The potential and the gauge coupling functions are our control functions whichallow us to modulate the IR dynamics. In [20] these were parameterized to be ofthe form V (Φ) = e−δΦ, and Z(Φ) = eγΦ respectively. It was argued that thesechoices allow for a range of locally critical IR behaviours as one scans over theparameters {γ,δ}. As we are interested not only in the IR behaviour, but also intranslating the local critical dynamics therein onto the ADS boundary, we need toensure that any such IR geometry can be patched to the asymptotically ADS region.We must therefore generalize the form of these functions. In doing so we chooseto fix the potential to have a Taylor expansion around the origin of field spaceof the form V (Φ) = Φ2 + · · · . This choice corresponds to an effective conformalmass term m2 = −2 for the scalar ensuring that we have simple fall-offs (withnon-normalizable and normalizable being z and z2 respectively asymptotically). Aconvenient choice for the functions which respects these constraints turns out to beV (Φ) =4υ2(1− cosh(υ Φ)) (6.2.3)Z(Φ) = eυΦwhere we have chosen to focus on a one-parameter family of theories, parametrizedby υ . This corresponds to the case γ + δ = 0 in the notation of [20]. We leaveexploration of the full parameter space for future study, and focus on this subsethenceforth.For computational simplicity we introduce inhomogeneity in our model bychoosing to break translational invariance in one-direction ofR2,1 as in [40, 73, 91].We construct a lattice in the x-direction while maintaining translational invariancein the y-direction.2 The x-translation breaking boundary conditions are imposed by2 We use the word “lattice” loosely for we choose not to impose any commensurability conditionsbetween the charge density and the unit cell.129choosing an inhomogeneous normalizable mode for the scalar field of the formΦ1(x) =C cos(kx) ,where C is the amplitude of the inhomogeneity and the k the wavenumber of thelattice.The manner in which this sourced inhomogeneity deforms the near horizongeometry from its homogeneous behaviour, and the effect that this has on the con-ductivity is the principal focus of this work. From the dual field theory point ofview this corresponds to determining how the UV parameters of the theory changethe trajectory of the renormalization group flow to create different phases of matterin the IR. As we explain below the presence of radially conserved quantities in thetheory mean that a great deal about the linear response functions, and therefore thephase of the dual field theory, may be extracted simply from knowledge of the nearhorizon geometry.Given these boundary conditions, a suitable metric ansatz for the investigationof these inhomogeneous phases is [73]:ds2 =1z2(− f (z)Qtt dt2+ Qzzf (z) dz2+Qxx(dx+ z2 Qxz dz)2+Qyy dy2)(6.2.4)with the functions Qab(z,x) depending both on z and x, thanks to the inhomogene-ity. In addition we introduce a redshift factor f (z) which, upon exploiting thescaling symmetries in the problem to fix the horizon to be at z = 1, can be chosento be:f (z) = (1− z)P(z) = (1− z)(1+ z+ z2−µ1 z32) (6.2.5)The factor µ1 can be thought of as a convenient parameterization of the tempera-ture, T ; they are related viaT =P(1)4pi=6−µ218pi. (6.2.6)This choice is useful as in the homogeneous limit the standard RN black hole maybe recovered by setting Qtt =Qzz =Qxx =Qyy = 1, Qxz =Φ= 0, A= (1−z)µ , andµ = µ1. We choose to measure all physical quantities relative to a fixed relation130between T and µ which we fix by taking µ = µ1 throughout. Also, for convenience,we choose A = (1− z)A0(z,x)dt ensuring thereby that the timelike component ofthe gauge field vanishes at the horizon.In order to ensure that our problem is well posed we must supply both consis-tent boundary conditions and an appropriate gauge condition to remove the gaugeredundancy of the Einstein equations. We choose to work in DeTurck (or har-monic) gauge – this is achieved by modifying the Einstein tensor via the additionof a new term involving the so-called DeTurck vector field ξ a:GHab = Gab−∇(aξb) (6.2.7)ξ a = gcd (Γacd(g)−Γacd(g¯))Here Gab is our original Einstein tensor, ξ a is the DeTurck vector, and GHab is themodified tensor appearing in the DeTurck equations. The DeTurck vector is de-fined using the difference in the Christoffel symbols, Γ, associated with our metricof interest, g, and a suitably chosen reference metric, g¯. The reference metricshould have the same asymptotic and conformal structure as the metric we are at-tempting to solve for. In our case we have found it convenient to use the RN metricas the reference metric. It can be shown that for the metric ansatz Equation 6.2.4,the DeTurck equations are elliptic [64] and therefore can be solved as a boundaryvalue problem.As we are interested in solving the original Einstein equations, we must ensurethat the DeTurck vector vanishes on-shell. Thus we choose boundary conditionssuch that the the DeTurck vector vanishes on the boundary. Provided the solutionsto our problem are unique and smoothly dependent on the choice of boundaryconditions, this should ensure the vanishing of the DeTurck vector. In practice, wecheck that the DeTurck vector is zero to a high numerical precision, so we may beconfident in the veracity of our solutions.After the gauge fixing procedure has been completed we obtain seven inde-pendent equations in the seven unknowns, (Qtt ,Qzz,Qxx,Qyy,Qxz,Φ,A0). It canbe checked that taking appropriate linear combinations of the equations decouplestheir principal parts, and each has an elliptic form.We next turn to the discussion of the boundary conditions. At each boundary131we require one boundary condition for each of the seven dynamic fields. We choosethe following conditions at the conformal boundary:Qtt(0,x) = Qzz(0,x) = Qxx(0,x) = Qyy(0,x) = 1, Qxz(0,x) = 0 (6.2.8)Φ′(0,x) =Φ1(x), A0(0,x) = µThe motivation for the metric boundary conditions is simply that one obtain ADSspacetime at z = 0. Labeling the Dirichlet boundary condition on A0(0,x) as µ isconsistent with its interpretation as the chemical potential in the dual field theory.The condition on the scalar field fixes the non-normalizable mode to be inhomoge-neous and thus sources inhomogeneity in the system.In the IR we must first impose the regularity of the black hole horizon. Thisis done by requiring Qtt(1,x) = Qzz(1,x) which ensures the surface gravity is con-stant along the horizon. The remaining boundary conditions may then be foundby expanding the equations of motion to leading order in (1− z). As two of theequation expansions are degenerate at this order we obtain the correct number ofhorizon boundary conditions. An intuitive understanding of these conditions canbe found by solving these leading order expansions for the radial derivatives of thefields at the horizon. We then see that these equations fix the radial derivatives ofthe fields in terms of their horizon values, and first and second spatial derivativesof the same fields. Therefore our horizon conditions consist of one Dirichlet condi-tion, necessary to define a regular horizon, and six (mixed) Robin-type conditions.At the spatial directions we impose that all fields are periodic, with periodL = 2pik . This allows for the sourced inhomogeneity to be a linear combinationof the basic harmonic cos(kx), and any of the higher harmonics cos(nkx) for anyinteger n. For example, [40] and, in a different context, [9, 62] have constructedsolutions with sourced inhomogeneity consisting of multiple modes with randomrelative phases in order to represent a ‘dirty’ lattice. While such investigations areinteresting we postpone the construction of such solutions and the investigation oftheir thermodynamic and transport properties to future work, and focus below onconstructing solutions sourced by the single harmonic cos(kx).1326.2.2 Perturbations and linear responseWe now turn to analysis of linearized perturbations, needed to extract informationabout the conductivity of the QFT in the linear response regime. We will primarilybe interested in computing the AC (optical) electrical conductivity as a function offrequency and temperature. The results have an intrinsic interest, and can also pro-vide a point of comparison to the DC conductivity formula which we will discussin Section 6.3. They thus provide a non-trivial check on our results. Combinedwith the DC conductivities, this provides a comprehensive picture of the behaviourof the theory at the temperature regimes we probe.The calculation of the conductivity in a holographic theory entails solving thelinearized perturbation equations derived from Equation 6.2.2. To derive these weperform the following expansion of the fields:gab = gˆab+ ε hab, Aa = Aˆa+ ε ba, Φ= Φˆ+ ε η (6.2.9)with hab, ba and η being the metric, gauge and scalar perturbations, respectively,and the hats indicating background fields. We work in leading order in ε and useas the background the solutions to Equation 6.2.2, subject to the boundary con-ditions described in Section 6.2.1. The symmetries of our background allow usto set by,hty,hzy,hxy to zero. Therefore the non-trivial components our metric andgauge perturbations are restricted to (htt ,hzz,hxx,hyy,htx,htz,hzx,bx,by,bz). In addi-tion since our background is static we may Fourier decompose the time dependenceof the perturbations:hab = h˜ab(z,x)e−iω t , ba = b˜a(z,x)e−iω t , η = η˜(z,x)e−iω t (6.2.10)As our equations are linear the time dependence is encapsulated in the factors ofthe frequency ω present in the equations.In analogy to the background equations we must impose appropriate gaugeconstraints on our problem in order to remove spurious degrees of freedom andobtain a match between the number of fields and independent equations of motion.In the case of the perturbation equations the gauge redundancy is more complexas, in addition to the diffeomorphism invariance of the gravity sector, the U(1)133invariance of the gauge sector must also be taken into account. In order to removethese gauge redundancies we choose to impose the deDonder and Lorentz gaugeconditions:τb = ∇a(hab−hgˆab2)= 0, χ = ∇aba = 0 (6.2.11)The procedure we use to impose the gauge conditions is very similar to that used toform the DeTurck equations in equation Equation 6.2.7. For example, in order toimpose the deDonder gauge we use a gauge fixing term constructed from a gaugetransformation, τb. Using the expression for the variation of the Ricci tensor andsubtracting the gauge fixing term ∇(aτb) = ∇(a∇chcb)− 12∇a∇bh gives us:δRab−∇(aτb) =−12∇c∇chab+12∇c∇ahcb+12∇c∇bhca−12∇b∇chca−12∇a∇chcb(6.2.12)=−12∇c∇chab+12[∇c,∇a]hcb+12[∇c,∇b]hcaThus we see that the principal part of the equation is hyperbolic and the goal ofthe gauge fixing procedure is achieved. Similarly, in the case of the gauge fieldequations we add a gauge fixing term of the form α(z,x)∇bχ . Here α(z,x) issome combination of the background fields which is determined by requiring thatthe principal part of each of the gauge equations takes the appropriate hyperbolicform.The end result of the gauge fixing procedure is 11 independent equations in 11dynamic fields whose principal parts are decoupled and of a hyperbolic form. Wecan now consider the boundary conditions. At the conformal boundary, these aresimple – we require bx to source the external external electric field and all other per-turbations to vanish suitably quickly such that asymptotically ADS is maintained.Therefore we choose the following:h˜tt(0,x) = 0, h˜zz(0,x) = 0, h˜xx(0,x) = 0, h˜yy(0,x) = 0, (6.2.13)h˜tx(0,x) = 0, h˜tz(0,x) = 0, h˜zx(0,x) = 0, b˜t(0,x) = 0,b˜z(0,x) = 0, b˜x(0,x) = 1134Here we have used the linearity of the equations to choose a convenient scale forthe magnitude of the external electric field.The situation at the horizon is more complicated. Physically our boundaryconditions must reflect the fact that all excitations are in-falling at the horizon.This is most readily observed by passing over to (the regular) ingoing coordi-nates Eddington-Finkelstein coordinates and ensuring that both the stress-energyand Einstein tensors are regular at the horizon. This will determine the leadingscalings of the fields at the horizon. Our results, which of course agree with thoseof [73], are:h˜tt(z,x) =P(z) h˜regtt (z,x), h˜yy(z,x) =P(z) h˜regyy (z,x) (6.2.14)h˜xx(z,x) =P(z) h˜regxx (z,x), h˜tx(z,x) =P(z) h˜regtx (z,x)h˜tz(z,x) =P(z)1− z h˜regtz (z,x), h˜xz(z,x) =P(z)1− z h˜regxz (z,x)h˜zz(z,x) =P(z)(1− z)2 h˜regzz (z,x), b˜z(z,x) =P(z)1− z b˜regz (z,x)b˜t(z,x) =P(z) b˜regt (z,x), b˜x(z,x) =P(z) b˜regx (z,x)η˜(z,x) =P(z) η˜reg(z,x)whereP(z) = (1− z)− iw4pi , w= ωT(6.2.15)captures the leading non-analytic behaviour of the fields near horizon and the re-maining analytic part is indicated by the superscript “reg”. Those regular fields maybe expanded in a power series in (1− z) near the horizon. Including the leadingand subleading orders in this expansion, one identifies 15 possible non-degenerateboundary conditions – 4 from the leading order and 11 from the subleading order.Only 11 of these, chosen for numerical convenience, are imposed. Consistency ofthe equations of motion demands that the remaining 4 conditions vanish on-shell.The vanishing of these constraints provides a non-trivial check on the accuracyof our numerical solutions. Finally, we will also demand that the solutions obeyappropriate periodicity conditions in the x-direction.135Once the solutions of the perturbation equations are available the AC conduc-tivity can be calculated as [40, 73] as:σ(ω,x) =jx(x)iω, with b˜x(z,x) = 1+ jx(x)z+O(z2) (6.2.16)From this the DC conductivity may be extracted by taking the ω → 0 limit of theRe(σ). We now turn to describing an alternative way of extracting the DC conduc-tivities, involving knowledge of the background solution alone.6.3 Analytic Expressions for the DC ConductivitiesWe now describe how to extract information regarding the electric, thermoelectric,and thermal conductivities from knowledge of the background fields alone. Thisdiscussion closely follows that of [40] (adapted to our coordinates).We begin by briefly reminding the reader of the form of the linear responsetransport equations in a 2+1 dimensional theory. At finite chemical potential, andtherefore finite density, the heat and electric currents may mix. Therefore Ohm’sLaw takes the more general form of:(JQ)=(σ αTα¯ T κ¯)(ET)(6.3.1)In our case J = Jx is the electric current and Q = T tx−µJx is the heat current andT=−∇xT is the thermal gradient. Our approach will be to perturb our holographicsystem such that E and T are introduced, consecutively, in the dual field theory.We will then find that certain radially conserved currents will allow us to read offthe conductivity pairs of (α¯,σ) and (κ¯,α) from the near horizon geometry of thebackground.6.3.1 Response from hydrodynamic perspectiveAt the outset however, we should remark the following. The DC response of thesystem comprises of two components: a response due to impurity scattering whichleads to momentum relaxation and Drude peak behaviour and a more primitivehydrodynamic response present even in translationally invariant systems. Specifi-136cally, we may write as in [61] the low frequency conductivity in the formσ(ω) =Kσ τ1− iω τ +σh(T,µ) , (6.3.2)to emphasize that there is a calculable contribution to transport even when momen-tum dissipation is swtiched off. Of course, this contribution has to be extractedafter subtracting off the divergent DC conductivity arising from the delta functioncontribution at ω = 0 in the τ → ∞ limit.In the hydrodynamic limit3 there is only a single response coefficient given byσh. To see this, note that the limit involves frequencies and momenta which aremuch smaller than the characteristic thermal scale. The lattice if present is treatedas a spatial long-wavelength perturbation about a homogeneous background. Thehydrodynamic energy-momentum and charge currents to first order in spatio-temporalgradients take the formT µν = ε uµ uν + pPµν −2η(T,µ)σµν −ζ (T,µ)ΘPµνJµ = ρ uµ +σh(T,µ)(Eµ −T Pµα∇α(µT)). (6.3.3)where Pab = gµν +uµ uν , is the spatial projector, σ µν is the shear tensor and Θ isthe fluid expansion. The thermodynamic data is encoded in the energy density ε ,pressure p and charge density ρ and we have assumed that the underlying systemis relativistic with Lorentz invariance being only broken by the choice of inertialframe picked by the fluid (through uµ ).4The shear and expansion contributions are tensor and scalar modes in the hy-drodynamic expansion, leaving the contribution coming from the electric field andthe gradient of the chemical potential and temperature to be the only vectorial partof transport. The existence of a single vectorial transport encoded in σh is relatedto the fact that fluids are required to satisfy the second law of thermodynamics.3 We use the phrase ‘hydrodynamic limit’ to refer to the low energy description of translation-ally invariant systems which is traditionally well described by relativistic hydrodynamics. Our aimhere is to illustrate the fact that underneath the Drude peak, there is a single frequency independent‘hydrodynamic conductivity’ captured by σh.4 In writing this expression we have chosen to fix some field redefinition ambiguity inherent inhydrodynamics by demanding that uµ be a timelike eigenvector of the stress tensor with eigenvaluebeing the energy density (Landau frame).137Allowing for three independent vector transport coefficients is inconsistent withthe existence of an entropy current with non-negative definite divergence, cf., [61].Translating this observation we require that in the hydrodynamic limit:5σ = σh , α = α¯ =−µT σh , κ¯ =−α µ =µ2Tσh (6.3.4)This can be used to check some aspects of numerics in the high temperature regime(where all our models exhibit metallic behaviour).6.3.2 Membrane paradigm for responseHaving understood what the relations we expect are, we can now turn to askingwhat the holographic modeling has to say about the transport coefficients of inter-est.We start by considering the electric and thermoelectric conductivities, σ , andα¯ . To extract the DC conductivities we modify the perturbation ansatz: (htt ,hzz,hxx,hyy,htx,htz,hzx,by,bz,η) are time-independent and bx = bx(z,x)− ε E t . We maythen proceed as described in Section 6.2.2 and derive the corresponding equationsof motion.6 It may then be easily checked that the linearized perturbation equationsfor the gauge field imply ∂x(√−gZ Fxr)= 0 and ∂r(√−gZ Frx)= 0, and therefore7J =√−gZ Fxr = constant. (6.3.5)As this quantity is a constant it can be evaluated anywhere including at thehorizon. In order to derive an explicit form for such an expression we wish toextract the leading scaling of the fields near the horizon. This may be done, asabove, by transforming into ingoing coordinates and demanding the regularity of5 A version of this relation was derived without invoking the second law but using holography in[57]. They however derive a relation that mixes zeroth and first order in gradients, which is at oddswith interpreting hydrodynamics as a low energy effective theory. Disentangling this leads to therelation we quote in Equation 6.3.4.6 For the purpose of the argument that follows the details of gauge-fixing of the equations isinessential.7 We will useJ andQ to denote the bulk conserved quantities, which will of course agree withthe boundary charge and thermal currents.138the stress-energy and Einstein tensor. The results are as follows:htt(z,x)' 4pih0tt(x)(1− z)+O(1− z)2, htz(z,x)' h0tz(x)+O(z−1)htx(z,x)'√Qxx(1,x)Qyy(1,x)h0tx(x)+O(1− z), hzx(z,x)'√Qxx(1,x)Qyy(1,x)4piT (1− z)h0rx(x)+O(1− z)hxx(z,x)' h0xx(x)+O(1− z), hyy(z,x)' h0yy(x)+O(1− z)η(z,x)' η0(x)+O(1− z), bt(z,x)' b0t (x)+O(1− z)bz(z,x)' 14piT (1− z)b0z (x)+O(1− z), bx(z,x)' log(4piT (1− z))b0x(x)+O(1− z)(6.3.6)and with the restrictions that:h0zx(x) = h0tx(x), h0zz(x) = 2h0tz(x)−h0tz(x), (6.3.7)b0x(x) =E4piT, b0z (x) = b0t (x)in order that Einstein and stress energy tensor be regular in this coordinate system.Note that for convenience we have chosen to include factors of the backgroundmetric fields evaluated at the horizon in our definition of h0tx,h0zx.Expanding our expression for J and utilizing these near horizon expansionswe obtain the following equation:√QxxQyy(JZ(φ)−A0 h0tx(x)Qzz)+b0′t (x)−E = 0 (6.3.8)where all background fields have been evaluated at the horizon.The next step is derive a similar expression for another conserved quantitywhich, as explained in [40], corresponds to the thermoelectric conductivity. To doso we note that if ξ is a Killing vector satisfyingLξF = 0 then we may define:G αβ = ∇αξ β +12ξ [αFβ ]σ Aσ +14(ψ−2θ)Fαβ (6.3.9)139withψ and θ defined asLξA= dψ and iξF = dθ .8 This has the important propertythat∇αG αβ = 3ξ β (6.3.10)Choosing the Killing vector, ξ = ∂t and utilizing the above identity and the equa-tions of motion it can be shown that ∇xG xr = ∂x(√−gG xr) = 0 and ∇rG rx =∂r(√−gG rx) = 0 and that therefore that9Q =√−gG rx = constant. (6.3.11)If we now choose that θ = −(−ε E x+A0(z,x)+ ε bt(z,x)) and ψ = ε E x, andexpandQ around the horizon as we didJ , we obtain at leading order:Q =−4pi T h0tx(x) . (6.3.12)This indicates that h0tx(x)= h0tx is a constant. Working at subleading order we obtaina constraint expression which, with a little rearranging, can be written as:√QxxQyy(h0tx2piT Qzz(pi T ∂z log(Qxx QzzQ3tt Qyy)−8pi T +6)− J A04pi T Qzz)+∂∂xh0tz(x)Qzz= 0(6.3.13)where again all background fields are being evaluated at the horizon. Integratingthis expression over one period of the background, the last term vanishes as a totalderivative whose boundary contributions cancel as a result of periodicity. We maythen solve for the only remaining perturbative field, h0tx, in terms ofJ . Perform-ing the same trick with equation Equation 6.3.8 we see that the ∂xb0t (x) disappearsunder integration, and the result gives us J =J (h0tx,E). Combining these tworesults we obtain an expression for the conductivity. To write the expressions com-8 Here L denotes the Lie derivative along the indicated field and iξ indicated contraction withthe vector field.9 The derivation of the conserved charges, J , and, Q, is presented in full generality in theappendix of [39]. We have also explicitly checked that the derivation of the conserved currents andcharges holds for the model we are considering.140pactly it is useful to introduce some notation; letI1 =∫dx(6−8pi T +pi T ∂z log(Qxx QzzQ3tt Qyy))√QxxQyy Q2zz,I2 =∫dx1Z(φ)√QxxQyy,I3 =∫dxA0Qzz√QxxQyy.(6.3.14)In terms of these integrals we find that the DC electric conductivity and the ther-moelectric conductivity are given asσ =2 I12 I1 I2− I23,α¯ =QT E=− 4pi I32 I1 I2− I23.(6.3.15)The calculation of the thermal and thermoelectric conjugate conductivities, κ¯ ,and α proceeds in a very similar fashion. In this case however our ansatz for theform of the perturbations contains two time-dependent components bx = bx(z,x)−ε t τ A(z,x) and htx = htx(z,x)− ε t Qtt(z,x) f (z)z2 .10 The restriction of regularity iningoing coordinates again determines the leading scalings for the fields. The resultfor the htx and bx fields is found to be:htx(z,x)'√QxxQyy(h0tx(x)+4piT hltx(x)(1− z) log(4piT (1− z)))+O(1− z)(6.3.16)bx(z,x)' b0x(x)+O(z−1)10 The form of the coupled metric and gauge field perturbation is chosen such that the equationsof motion remain time-independent. In writing the form of the metric perturbation as above wehave tacitly modified our background metric ansatz such that Qxz = 0. This is done to avoid themore complicated form of the metric perturbation necessary if Qxz 6= 0. It has been checked that theresponse function results derived from the near horizon behaviour do not change as a result of thismodification.141while all other fields exhibit the same scalings and constraints as displayed in equa-tions Equation 6.3.6. The constraints displayed in Equation 6.3.7 are modified bythe presence of a logarithmic term in the htx expansion and the lack of an externalelectric field. The resulting constraints are given by:h0zx(x) = h0tx(x), h0zz(x) = 2h0tz(x)−h0tz(x) (6.3.17)b0z (x) = b0t (x), hltx =−Qttτ4piT√QzzQxxIn order to obtain expressions for the response functions which do not involvethe background fields we expand the conserved quantities of,J , andQ to leadingand subleading order, respectively, near the black hole horizon. The leading orderterm in the expansion ofQ again indicates that h0tx(x) is:Q =−4pi T h0tx(x) (6.3.18)While the remaining two constraints displayed in equation Equation 6.3.19 may beintegrated over a single period such that they become linear equations inJ h0tx andτ .√QxxQyy(JZ(φ)−A0 h0txQzz)+b0′t (x) = 0√QxxQyy(h0tx2piT Qzz(piT∂z log(QxxQzzQ3ttQyy)−8piT +6)− J A04piT Qzz)+∂∂xh0tzQzz+ τ = 0(6.3.19)This information is sufficient to calculate the thermal and conjugate thermo-electricconductivities:κ¯ =QT τ=16pi2T I22 I1 I2− I23α =JT τ=− 4pi I32 I1 I2− I23(6.3.20)Once we have an expression for κ¯ we may easily calculate the conjugate ther-142mal conductivity, κ which corresponds to the the thermal conductivity at zero elec-tric current. This is done via the relationκ = κ¯− α2Tσ=8pi2TI1(6.3.21)It is reassuring to note that the α = α¯ as it should since our system does notbreak time reversal invariance. We may also check that the high temperature limitof the response functions behaves in the expected fashion. This limit may be ex-tracted by expanding around Tµ =T√6−8piT →∞. In this limit the solution resemblesthat of a Schwarzschild-ADS black hole and so appropriate field substitutions are:A0→ δε µ, Qzz→ 1+δε ∆Qzz, Qtt → 1+δε ∆Qtt , (6.3.22)Qxx→ 1+δε ∆Qxx , Qyy→ 1+δε ∆Qyy Qxz→ δε ∆Qzz,φ → δε ∆φ , µ → δε µwhere we may now expand around δε = 0.11 We find in this limit that α = α¯ → 0and κ¯ → ∞ diverges as the effects of momentum dissipation are removed from thesystem. The electrical conductivity, σ , asymptotes to Z(0) = 1, the known resultfor Schwarzschild-ADS. Furthermore, we can confirm that in the hydrodynamicregime the relation between the conductivities Equation 6.3.4 is satisfied. In fact,from Equation 6.3.15 and Equation 6.3.20 we learn that in the hydrodynamic limitthe background geometry should satisfy 2 I1 I2− I23 = 0.12It was shown in [73] and [39] that the breaking of translational invariancein holographic models produces a low frequency AC conductivity well fit by theDrude model of conductivity. We confirm that this is true in our model in the ap-pendix. It will therefore be interesting to test if our model obeys the Wiedemann-Franz law at any point in its phase diagram. We therefore calculate the Lorenz11 The high temperature limit may also be accessed by undoing the scalings of the action and equa-tions of motion which are used to keep the location of the horizon fixed at z= 1 as the temperature ischanged. As the location of the black hole horizon, zp, then approaches the conformal boundary asTµ → ∞ we may find the high temperature behaviour of the response functions by expanding aroundzp = 0. The appropriate expansions of the fields in this limit is determined by their known conformalbehaviour. As expected the results from this approach agree with those presented in the text.12 Strictly speaking this relation is valid in the translationally invariant case, where the DC con-ductivity contribution of the Drude peak diverges.143factors for both κ and κ¯ as follows:L¯ =8pi2 I2I1(6.3.23a)L =4pi2 (2 I1 I2− I23 )I21(6.3.23b)In Section 6.4 we will see that the that these expressions are not constant in eitherthe metallic or insulating phases of our model, confirming departures from theWiedemann-Franz law.6.4 Transport Results for Holographic SystemsWe regard the present effort as a first step of exploring the vast phase diagram ofthe the effective holographic theories, identifying interesting corners for furtherstudy. In this section we discuss features of the DC conductivities and the opticalconductivity, and qualitative changes in the physics as we vary parameters. Weargue that these changes indicate the existence of metal-insulator quantum phasetransitions at various loci in the phase diagram.13Let us start with a brief description of the numerical techniques we employ, be-fore describing our main results. For those interested, much more detail regardingthe development and testing of our numerical methods is provided in Section D.1.In order to solve our PDEs numerically we discretize them using spectral methods[17, 126]. A Chebyshev grid was employed in the radial direction and a Fouriergrid in the spatial direction, which imposes spatial periodicity, as discussed above.The solution of the non-linear background equations uses both Newton and quasi-Newton methods, whereas the solution of the perturbation equation only necessi-tates the inversion of a matrix, for which we use direct methods.6.4.1 DC conductivitiesWe start by exploring the direct electric and thermoelectric conductivities. Theseare calculated using horizon data, as explained in Section 6.3. The electric con-13 The phrase, phase diagram, here refers to changing both the sources in a given theory (by tuningthe period of the lattice set by k) as well as explorations across theories (by changing Lagrangianparameter υ).144ductivity is also calculated as the zero frequency limit of the optical conductivity.Besides providing a check of the numerics, the low frequency behaviour of theoptical conductivity helps in understanding and elucidating the IR physics.Since we are working at finite temperature, we cannot probe the metal-insulatorquantum phase transition directly. Nevertheless, by interpolation of our knowl-edge of both the temperature dependence of the DC and optical conductivities tosufficiently low temperatures, we can diagnose the presence of such transition asfunction of parameters. In our exploration we fix C = 1.5, and discuss the phasediagram as function of υ as well as the perturbation wavenumber k. Varying thesetwo parameters we find both metallic and insulating regimes, and transitions be-tween them.In Figure 6.1 we show a representative sample of the DC conductivities withk = 1 and varying υ . We clearly observe the transition from an insulating to metal-lic behaviour as the value of υ is increased. For low value of υ , which includes theEinstein-Maxwell model of [73], we find a distinct insulating behaviour: the con-ductivity decreases at low temperatures and seems to vanish at zero temperature.On the other hand, for sufficiently large υ the opposite behaviour is manifest: theconductivity is increasing with temperature and seems to diverge at zero tempera-ture. In the transition region, the conductivity shows no distinct trend – this is theregion which is a bad or incoherent metal at low temperatures.A similar trend can be seen in the DC thermoelectric conductivity, which is alsomonotonically increasing as we lower the temperature, for small values of υ . As weincrease υ the curve begins to kink downwards at low temperatures until eventuallya well defined turning point is formed. This turning point migrates towards largertemperatures as we continue to increase our control parameter. We conclude thatthe transition between metallic and insulating behaviour exists in this observableas well.It is interesting to note that no such transitions occur in the thermal conductiv-ity, which displays a simple monotonic increase as a function of temperature for allvalues of parameters we examined. Thus our theories are all good thermal conduc-tors. In some sense this is not surprising as the Wiedemann-Franz law is explicitlyviolated in our expressions for Lorenz factors in equations Equation 6.3.23a andEquation 6.3.23b. This is confirmed by the numeric results presented in Figure 6.3.1450 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.016810121416182022DC conductivity as a function of TT/µ σ  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.070 0.01 0.02 0.03 0.04 0.05 0.06−26−24−22−20−18−16−14−12−10−8DC thermoelectric conductivity as a function of TT/µα/µ  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07Figure 6.1: Plots of the DC electrical and thermoelectric conductivitiesagainst temperature for various theories parameterized by υ , with C =1.5,k = 1 held fixed. We clearly see the existence of the metallic andinsulating regimes separated by an intermediate region.1460 0.01 0.02 0.03 0.04 0.05 0.06024681012DC thermal conductivity as a function of TT/µκ µ  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.070 0.01 0.02 0.03 0.04 0.05 0.0600.20.40.60.811.21.4DC thermal conductivity at                zero electrical current as a function of TT/µκ/µ  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07Figure 6.2: Plots of both forms of the thermal conductivity, κ and κ¯ as a func-tion of Tµ . We note both the qualitative similarity of the two quantitiesand the insensitivity to the variation in the υ parameter. In all casesthat we have examined, including those associated with the data usedto construct Figure 6.4, the thermal conductivities were seen to increasemonotonically with temperature.1470 0.01 0.02 0.03 0.04 0.05 0.0668101214161820222426Lorenz factor as a function of TT/µL  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.070 0.01 0.02 0.03 0.04 0.05 0.0600.511.522.53Lorenz factor at                          zero electrical current as a function of TT/µL  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07Figure 6.3: Plots of the Lorenz factors associated with L and L¯ as a functionof Tµ . The fact that these factors are neither constant as a function oftemperature nor υ indicates that the Wiedemann-Franz law is violated.This is in accord with the lack of a phase transition in the thermal con-ductivity. 1480.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.60.70.80.911.11.2Metal−insulator transition of DC conductivtiyνkFigure 6.4: The phase plot in the (υ ,k) plane illustrating the metal insula-tor transition. The green data points correspond to regimes which areclearly metallic in character with monotonically increasing DC electri-cal conductivity at low temperatures. Likewise the red data points corre-spond to insulating phases characterized by a monotonically decreasingDC electrical conductivity at low temperatures. The intermediate regionbetween these corresponds to the transition region where a turning pointis still evident in the profile at low temperatures.6.4.2 Metal-insulator transitionsWe have seen that the finite temperature results lend themselves to interpretationas indicative of a zero temperature metal-insulator transition. We characterize sucha phase transition by a qualitative change in the low temperature behaviour of theresponse functions. One can delineate regions of the phase diagram where theholographic theories we consider describes good metals, bad metals and insulators.We now search for such transitions as function of υ and k.The result, displayed in Figure 6.4, separates the parameter space into theregimes of clear metallic and insulator phases, separated by intermediate regimes149in which the conductivity is neither monotonically increasing or decreasing. Wesee that increasing υ reduces the relevance of the sourced inhomogeneity, such thatthe transition to a metallic phase occurs at lower values of k.The intermediate phase which straddles the phase transition region can beviewed as one where the competition between metallic and insulating orders isstrong. It is interesting to speculate in analogy with the domain model for mag-netic phase transitions, that one is encountering pockets of the competing phases.This leads to incoherence in the transport, reducing for instance the conductivityfrom its metallic value.We note that the phase separation between metallic and insulating behaviouroccurs along a locus k?(υ) which is monotone decreasing – as we increase the lat-tice wavelength, we encounter a transition at lower values of υ . This suggests thatthe efficacy in translating the lattice between UV and IR regions in the geometry isplaying a role in the presence/absence of “charge carriers”.14 The DC conductivityis effectively a proxy for the weight of the charge carrier spectral function at van-ishing frequency. The support of this spectral function is localized in the vicinityof the horizon. This immediately follows from the fact that we have a membraneparadigmesque formula for the conductivity.Finally, we specialize to υ = 0 which describes the Einstein-Maxwell modelof [73]. This allows us to probe the existence of a phase transition as a functionof the wavenumber k alone.15 The results for the DC electrical conductivity aredisplayed in Figure 6.6 and clearly show the existence of a quantitative change inthe temperature dependence as k is varied. However at temperatures which we mayreliable access the phase is always insulating for our choice of parameters (withC = 1.5 held fixed). This is in contrast to the metallic phases observed for theparameters chosen in [73].16 We have seen evidence that the transition may occurat lower temperatures however these are difficult to probe reliably. The qualitativeshift in behaviour when moving from finite υ to υ = 0 can be illustrated by directexamination of the background field solutions as seen in Figure 6.5.14 We use the phrase “charge carriers” somewhat loosely since we are talking about transport in astrongly interacting system with no obvious quasiparticles.15 Metal-Insulator transitions in Einstein-Maxwell theory deformed by helical lattices were dis-cussed in [43].16 We have checked that for the parameters C,k chosen in [73] we also encounter metallic phases.150T=0.0015, k=2.02, υ=0.00200.510123024Gauge Potential00.510123−4−20Neutral Scalar00.510123012Qtt00.510123012Qzz00.510123012Qxx00.510123012Qyy00.510123−202Real QxzT=0.0015, k=2.02, υ=000.510123234Gauge Potential00.510123−0.500.5Neutral Scalar00.5101230.511.5Qtt00.5101230.511.5Qzz00.5101230.60.81Qxx00.51012300.51Qyy00.510123−0.0500.05Real QxzFigure 6.5: Setting υ to be strictly zero means that the IR evolution of thescalar field flattens out. As this behaviour of the scalar controls manyaspects of the IR physics qualitative changes in the response functionsare to be expected. 1510.5 1 1.5 2 2.5 3 3.5 4x 10−30.50.60.70.80.911.11.2Metal−Insulator phase as a function of kT/µσ  k=0.71k=0.77k=0.83k=0.89k=0.95k=1.01k=1.07k=1.13k=1.19k=1.25k=1.31k=1.37k=1.43k=1.49k=1.59k=1.65k=1.71k=1.77k=1.83k=4.01Figure 6.6: The DC conductivity in the limit υ = 0 for different latticeswavenumbers,k. We notice that for lower values of the wavelength kthe conductivity is decreasing at low temperatures. As we increase k theDC conductivity begins to flattens at lower temperatures though for thischoice of amplitude, C = 1.5, for the sourced inhomogeneity the con-ductivity never transitions into the metallic phase for the temperatureswhich we can reliably access.6.4.3 Optical conductivityIt is interesting to examine the AC electrical conductivity in the vicinity of thetransition between good metals and insulators. We observe that the influence of theυ parameter on the profile of the real and imaginary parts of the AC conductivityis minimal until Tµ is sufficiently small. This is in keeping with the profiles ofthe DC electric conductivity shown in Figure 6.1, where the profiles do not beginto strongly differentiate until values of Tµ ' 0.06 are reached. At lower values ofTµ the characteristic profiles associated with Drude behaviour begin to acquire adistinct spread as a function of υ . This AC conductivity of manifestation of themetal-insulator transition, cf., Figure 6.7 and Figure 6.8 .Our testing the zero frequency limit of the optical conductivity, agrees cleanlywith the explicit evaluation of the DC conductivity using the membrane paradigmformulae. In the real part of σ(w) we observe the characteristic Drude peak. Wecan confirm that the intercept σ(w= 0) agrees with the DC result obtained from the152membrane paradigm, providing us with a nice consistency check of the numerics.Inspection of the low frequency optical conductivity lends further evidence toour physical picture of the low temperature transport. In Figure 6.9 and Figure 6.10we confirm that as the temperature is lowered for parameter choice in the insulatingphase spectral weight is shifted from lower to higher frequencies, as expected.We also check for potential mid-IR scaling regimes. A convenient way to iden-tify this behaviour is using the diagnostic quantity which we label F(w) (cf., [40]):F(w) = 1+w|σ(w)|′′|σ(w)|′ (6.4.1)If |σ(w)| develops a scaling regime such that it behaves as |σ(w)| ∼C1 +C2wν ,the quantity F(w) will be equal to a constant given by the scaling exponent ν .In Figure 6.11 and Figure 6.12, and Figure 6.13 and Figure 6.14, we plot F(w)for a variety of values of Tµ and υ , including those displayed in Figure 6.7 andFigure 6.8. We observe that as Tµ is decreased the profiles of the diagnostics F(w)begin to flatten out and the existence of a scaling regime becomes a real possibility.However the existence of such a scaling regime requires the fine tuning of υ (orTµ ) to a very narrow parameter range. This ranges questions regarding the genericnature and therefore physical significance of the behaviour. However, even whenscaling regimes do exist, the scaling exponent seem to be generically different than2/3. Further questions regarding the significance and robustness of this scalingregime is postponed to future work.176.4.4 High temperature limitAs described in Section 6.3 the form of the response functions should exhibit hightemperature behaviour consistent with that of the Schwarzschild-ADS black hole.In this limit σ → 1, α = α¯ → 0 and κ¯ → ∞. Therefore both the metallic and insu-lating low temperature phases must transition to that generic behaviour, determinedby the conformal invariance of the UV theory, as the temperature is increased.An interesting point to note is that, as seen in Figure 6.1, the DC electricalconductivity increases to values well above unity in the insulating phase. This can17 For a discussion of the difficulties involved in working in this frequency regime please seeSection D.2.1530 0.5 1 1.5 2 2.5 3234567891011Real conductivity at T/µ=0.015949w/TRe(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 0.5 1 1.5 2 2.5 30.511.522.533.544.555.5Imaginary conductivity at T/µ=0.015949w/TIm(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 0.5 1 1.5 2 2.5 3456789101112Real conductivity at T/µ=0.012167w/TRe(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure 6.7: We examine the effect of the control parameter υ on the AC con-ductivity as Tµ is lowered. As the value ofTµ is lowered the influence ofυ on the form of the real conductivity curves becomes more apparent.In this limit the curves for different values of υ can be seen to clearlydifferentiate at lower values of w.1540 0.5 1 1.5 2 2.5 30123456Imaginary conductivity at T/µ=0.012167w/TIm(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 0.5 1 1.5 2 2.5 356789101112Real conductivity at T/µ=0.0085301w/TRe(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 0.5 1 1.5 2 2.5 301234567Imaginary conductivity at T/µ=0.0085301w/TIm(σ)  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure 6.8: A continuation of the sequence of figures begun in Figure 6.71550.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.29.49.69.81010.210.410.6Real conductivity at ν=0.01w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.29.49.69.81010.210.410.610.811Real conductivity at ν=0.03w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.29.49.69.81010.210.410.610.811Real conductivity at ν=0.05w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure 6.9: Examining the real part of the AC conductivity as a function ofωT for a variety ofTµ and υ . At small υ the constantTµ curves interceptand overlap for lower values of w. This can be interpreted as the ACrepresentation of the incoherent phase. As υ is increased the constantTµ curves separate and differentiate as we transition into a conductingphase.1560.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.29.49.69.81010.210.410.610.811Real conductivity at ν=0.06w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.29.49.69.81010.210.410.610.811Real conductivity at ν=0.07w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.599.51010.51111.512Real conductivity at ν=0.1w/TRe(σ)  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure 6.10: A continuation of the sequence of figures begun in Figure 6.91575 5.5 6 6.5 7 7.5 82.752.82.852.9Mid IR scaling test at T/µ=0.019889w/TMid−IR scaling test  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.15 5.5 6 6.5 7 7.5 82.82.852.92.95Mid IR scaling test at T/µ=0.015949w/TMid−IR scaling test  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure 6.11: The behaviour of the diagnostic function F(w) as we scan formid-range scaling behaviour as a function of υ . At low temperatures,and for appropriately chosen values of υ , the existence of a scalingregime is possible. For example, at temperature of Tµ = 0.01267 weobserve that the our test function flattens out at υ ' 0.09.1585 5.5 6 6.5 7 7.5 82.752.82.852.92.953Mid IR scaling test at T/µ=0.012167w/TMid−IR scaling test  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.15 5.5 6 6.5 7 7.5 82.652.72.752.82.852.92.9533.05Mid IR scaling test at T/µ=0.0085301w/TMid−IR scaling test  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure 6.12: A continuation of the sequence of figures begun in Figure 6.11be understood as consequence of the sum rule obeyed by the optical conductivity –the suppressed low frequency spectral weight in the insulating phase has to transferto high frequencies, which are still low compared to the scale of the chemicalpotential. Therefore for those insulating phases, a turning point must exist whenthe sign of the slope of the DC electric conductivity reverses, as the low temperaturephysics begins to transition to the generic high temperature behaviour. Conversely,the existence of such turning point could be another indication of the transition toan insulating phase.1595 5.5 6 6.5 7 7.5 82.652.72.752.82.852.92.953Mid IR scaling test at ν=0.1w/TMid−IR scaling test  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240035 5.5 6 6.5 7 7.5 82.652.72.752.82.852.92.953Mid IR scaling test at ν=0.09w/TMid−IR scaling test  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure 6.13: The behaviour of the diagnostic function F(w) as we scan formid-range scaling behaviour as a function of temperature. The reversalof the slopes of F(w) as a function of temperature, for a fixed υ , implythat it should be possible to tune to a scaling regime by appropriatelyspecifying the temperature.1605 5.5 6 6.5 7 7.5 82.652.72.752.8Mid IR scaling test at ν=0.02w/TMid−IR scaling test  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240035 5.5 6 6.5 7 7.5 82.652.72.752.8Mid IR scaling test at ν=0.01w/TMid−IR scaling test  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure 6.14: A continuation of the sequence of figures begun in Figure 6.13In Figure 6.15 we plot the values of Γcrit = Tµ at the turning point versus υand k. For the DC electrical conductivity it can be seen that the position of thisturning point decreases steadily as one moves from the insulating regime towardsthe metallic transition. We also plot the corresponding turning point in the DCthermoelectric conductivity (right panel). It should be noted however that a further,higher temperature, change of slope must exist for the thermoelectric conductivity,as it is expected to go to zero in the high temperature limit.1610.020.040.060.080.10.60.811.200.020.040.060.080.1kTurning point of the DC electric conductivityνT/µ0.020.040.060.080.111.051.11.151.20.010.0150.020.0250.030.0350.040.0450.05νTurning point of the DC thermoelectric conductivitykT/µFigure 6.15: The DC electric conductivity starts off at a finite value of Γcrit =Tµ in the insulating phase and decreases rapidly as we approach themetal-insulator transition. On the other hand, in the thermoelectricconductivity, the existence of a turning point sets in as we enter themetallic phase and steadily increases in Tµ as we tune υ and k to movedeeper into the metallic phase.1626.5 Conclusions and OutlookThe main goal of this article has been a preliminary exploration of the large phasespace of effective holographic theories, with a particular focus on understand-ing the efficiency of low energy transport in these models. More specificallywe focused on the low frequency conductivity in a phenomenologically moti-vated ADS/CFT set-up and examined the predilection of the system towards metal-insulator phase transitions. We view our investigation as offering a large set of toymodels, with features that can be dramatically different from those of the muchstudied Einstein-Maxwell theory. Our analysis has identified interesting loci in thephase diagram where qualitative changes of the IR physics take place. The naturalnext step is taking a closer look at these locations to get a better understanding ofthe physics that drives these transitions in holographic systems.One of the interesting regimes is that of incoherent, or bad metals which lie atthe interface of the metal-insulator transition. A conjecture was put forward in [56],that transport in such incoherent metals was to be governed by diffusion processes.The models studied here provide a natural testing ground for this conjecture.Additionally, it is interesting to study the low lying quasinormal modes in theincoherent and insulating phases. The motion of those quasinormal modes in thecomplex plane can often elucidate the dominant physics governing the transitionor cross-overs in the qualitative behaviour of the conductivity. In this context, itwould also be interesting to quantify the scaling of the spectral weight and DCconductivity at low frequencies. In particular, examination of the residue of thelightest quasinormal mode (the hydrodynamic mode that contributes to charge dif-fusion), should illuminate whether the reduction in the conductivity as we enterthe regime of incoherent metals (from the metallic side) is caused by the drop inspectral weight or if some other physics is responsible.From a gravitational viewpoint it would be useful to know if there is a char-acteristic feature of the near-horizon geometry which results in this phenomenon.Our preliminary investigations were inconclusive in ascertaining a sharp feature ofthe near-horizon geometry, which could be held responsible for the incoherencein charge transport. Ideally one would conjure up a geometric observable that issensitive to transport. What we can definitely conclude is that for fixed υ varying163k demonstrates clear changes in the relevance of the inhomogeneity in all the met-ric components. Since decreasing k increases the relevance of the inhomogeneousscalar source, this makes it clear that smaller values of k will have a more strongerimpact in the IR transport. Comparing across values of υ is of course more com-plicated, since we exploring behaviour in the space of theories. Indeed even in thehomogeneous case we see differing IR behaviour as we tune υ .We have further seen that in certain small regimes of our parameter space in-termediate frequency scaling may be possible. While our analysis of this effect hasnot been comprehensive, it appears that we require a certain amount of fine-tuningin order to achieve scaling behaviour. Moreover, in most of the cases we looked atthe exponent ν was not 2/3, which was the value seen for the cuprates [127] and inυ = 0 models [72, 73]. 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SL(2,Z) action on three-dimensional conformal field theorieswith Abelian symmetry. 2003.174Appendix AAppendix: Holographic HiggsPhasesA.1 Vortex Solutions in Flat SpacetimeThe vortex solutions for the flat space Abelian Higgs model are well known. Herewe provide a brief discussion following [125], for further information see also[110].The action of the Abelian Higgs model is:L =−14FµνFµν +12(∇µΨ)∗(∇µΨ)− 14λ (|ψ|2−F2)2The constant F is proportional to the VEV of the charged scalar field Ψ breakingthe U(1) gauge symmetry. Finite energy configurations of the fields require thatthey obey the asymptotic conditions:|Ψ| → ψ0, ∇µΨ= (∂µΨ− iqAµΨ)→ 0, as x→ ∞Requiring cylindrical symmetry, the form of the scalar and gauge fields at infinity175are constrained to be:Ψ(r,θ)→ ψ0 exp(iα(θ))Aµ →− iqδµψψ=1qrdαdθ, as r→ ∞The winding of the phase, α , at infinity is an integer, s, which is related to thequantized magnetic flux through the plane orthogonal to the magnetic vortices:s =q2pi∫ 2pi0Aθ rdθ =q2pi∮A.dl =q2pi×magnetic fluxIn order to see explicitly the localized nature of these vortices one is required toexamine the equations of motion. Using the ansatz Ar = A0 = 0, Aθ = A(r) andΨ(r,θ) = ψ(r)exp(isθ) we obtain the following equations:−ψ(r)[( sr−qA(r))2+λ (ψ(r)2−F2)]+ψ ′(r)r+ψ ′′(r) = 0ψ(r)2(sqr−q2A(r))− A(r)r2+A′(r)r+A′′(r) = 0The falloff conditions imply that ψ(r)→ψ0 and A(r)→ sqr as r→∞. We may usethis information to linearize the Maxwell equation for large r by setting ψ(r) =ψ0.Solving the resulting equation yields:A(r)−−−→r→∞sqr+C1√rexp(−qFr)As the scalar and gauge field approach their asymptotic configurations at large r,we expect the action to be dominated by the potential term. Therefore in order tofind the asymptotic behavior of the scalar field we examine perturbations of thepotential. Requiring that the first derivative of the potential to vanish fixes theminimum at ψ0 = F2 . The fluctuations around this are of the form F2λρ(r) whereρ(r) is the deviation of the scalar field from ψ0. Using this approximation for the176potential in the scalar equation and setting A(r) =sqrwe obtain:ψ(r)−−−→r→∞ ψ0+C2 exp(−√λFr)The localized nature of the vortex is evident from the exponential decay of thefields to their asymptotic values for large r. The full solution can be obtained bysolving the equations numerically.A.2 Details of the NumericsWe solve the equations with the boundary conditions listed in section Section 2.4.2numerically using successive over-relaxation SOR algorithm. To this end we dis-cretize the equation on a lattice of finite mesh-size h covering the domain of inte-gration, such that continuous spatial coordinates (ρ,w) are represented by discretepairs (wi,ρ j), where 1 ≤ i ≤ Nw,1 ≤ j ≤ Nρ are integers. We use a second or-der finite differencing approximation (FDA), where the derivatives are replacedwith their finite differencing counterparts, e.g. ∂wR→ (Ri+1, j−Ri−1, j)/2h,∂ρR→(Ri, j+1−Ri, j+1)/2h etc. Following discretization, we thus obtain finite differenceequations, at every mesh point, for each field. We iteratively solve the entire sys-tem of algebraic equations using pointwise SOR starting with an initial guess forthe fields, until a desired precision is achieved. Typically we initialize our scalarand A0 gauge field with the values of the homogeneous solution (found by solvingthe ODEs using shooting) and set the Aθ field to its expected asymptotic value ofs/q. Along the horizon an initial guess is made for the scalar and Aθ fields whichinterpolates exponentially between the zero boundary condition at ρ = 0 and theexpected asymptotic values at the ρ → ∞ boundary. We use similar SOR parame-ters for all fields. These are calculated at each step via Chebyshev iteration. In thisiteration the spectral radius of the Jacobi iteration is chosen, for simplicity, to bethat of the Laplace equation with Dirichlet boundary conditions, see Section 19.5of [109] for further details and the algorithm.While Dirichlet boundary conditions are implemented by assigning the fieldstheir initial values throughout the relaxation procedure, Neumann or Robin bound-177ary conditions are updated after each iteration. We do this by using the backwardsFDA derivative operators to update the boundary grid points based on the valuescalculated at the interior points. It turns out that at the horizon such a straight-forward implementation of the regularity conditions Equation 2.4.8 is numericallyunstable. As these conditions relate the radial and tangential derivatives of thefields along the horizon they yield, upon discretization, a pair of coupled polyno-mial equations which relate the values of the fields at grid points in the near horizonregion. Attempting to solve these polynomial equations to update the values of theboundary grid points after each iteration resulted in instabilities, which we attributeto the fact that the linearized scalar equation near the horizon is ill-posed (the effec-tive mass terms and the elliptic operator have the same sign). The physical reasonof this instability can be traced to the fact that the effective scalar mass in the nearhorizon region violates the Breitenlohner-Freedman bound, so that it triggers aninstability and formation of a condensate.We found that a stable implementation of the constraint equations Equation 2.4.8is to evaluate all terms in the equation, except for the radial derivative, on the lineof grid points just before the horizon, and to then use the FDA form of the radialderivative to extrapolate to the values of the fields on the horizon. This approachis consistent with the bulk equations of motion and identical to implementing thedesired constraint equations when the continuum limit is taken (i.e. the limit inwhich the step size is taken to zero).While our numerical lattice extends all the way from the horizon w = 1 to theconformal boundary w = 0, it covers only finite domain in the transverse direction0< ρ < ρcut . The truncation radius ρcut is chosen such that our numerical solutionsare altered by less than 0.01% when ρcut is increased. Typically we use ρcut ∼100− 120. In addition, we checked that asymptotically our PDE solutions of thevortex configuration converged to the ODE solutions of the translationally invariantconfiguration at the transverse boundary to accuracies of 0.01% or higher.Finally we discuss convergence of our finite-differencing numerical solutions.The rate of convergence is assessed based on the assumption that in the continuumlimit, when the grid-size tends to zero, the discrete solution on the mesh h, des-ignated uh, approaches the continuum solution, u∗, namely uh = u∗+O(hn). Thepower n measures the rate of convergence. It can be calculated by running sim-178ulations with similar parameter settings on a sequence of meshes with decreasingmesh-spacings h,h/2 and h/4, and computing n = log2(uh− uh/2)/(uh/2− uh/4).We found that the convergence rate in our case is very close to n = 2 for the scalarand Aθ fields as expected for second order FDA, provided the numerical lattice issufficiently dense to ensure we are in convergent regime. The convergence rate forfor the A0 field was seen to be somewhat lower at, n' 0.7. Typical meshes that useto obtain results seen in Figure 2.7 are of size Nw×Nρ = 400×1000, which yieldsgrid spacings of order hw×hρ ' 0.0025×0.1.179Appendix BAppendix: Striped Order inAdS/CFTB.1 Asymptotic ChargesB.1.1 Deriving the chargesSince our ansatz is inhomogeneous and includes off-diagonal terms in the met-ric, and our action is not standard (in that it includes the axion coupling) we havere-derived the expressions for the charges and other observables in our geome-try. In deriving the asymptotic charges of our spacetime, for the four dimensionalEinstein-Maxwell-Higgs theory we discuss in the main text, we follow the covari-ant treatment of [107, 108]. We refer the reader to those papers for details of themethod used.The bulk action has to be supplemented by boundary terms of two types. First,there are boundary terms needed to ensure that the variational problem is well-defined. Then there are counter-terms, terms depending only on the boundary val-ues (leading non-normalizable modes) of fields on the cutoff surface, which areadded to render the on-shell action and the conserved charges finite. Both kinds ofboundary terms are the standard ones for Einstein-Maxwell-Higgs theory; the ad-ditional axion coupling does not necessitate an additional boundary terms of either180kind as long as the scalar mass satisfies m2 < 0.We find it convenient to study the first variation of the on-shell action, whichalways reduces to boundary terms. The expression for the regulated first variationof the on-shell action can be differentiated with respect to the boundary values ofthe bulk fields, to give finite expressions for the conserved charges. We write thoseexpressions below in terms of the asymptotic expansion of the fields occurring inour ansatz, carefully taking into account the differences between our coordinatesystem and the standard Fefferman-Graham form of the asymptotic metric, whichis used to derive the standard expressions in the literature.Having explained our procedure, we now display the expressions for the ob-servables used in the main text. We first assume the radial coordinate is in thestandard Fefferman-Graham form, and then discuss additional terms arising fromchange of coordinate necessary to bring our asymptotic metric into the standardform.For the scalar fields ψ , one can write asymptoticallyψ(x,r) = ψ(0)(x)r−λ−+ψ(1)(x)r−λ+ (B.1.1)withλ± =32±√94+m2. (B.1.2)We set ψ(0)(x) = 0 as part of our boundary conditions, then the coefficient ψ(1)(x)is the spatially modulated VEV of the scalar operator dual to ψ .Similarly, the gauge field can be expanded near the boundary asAµ(x,r) = A(0)µ (x)−A(1)µ (x)r. (B.1.3)The functions A(1)µ (x) correspond to the charge and current density of the boundarytheory.As for the boundary energy-momentum tensor, the expression is fairly simplein odd number of boundary dimensions, and we have checked that it is not modified181by the matter action. With our normalization convention one can writeTi j = 6g(3)i j , (B.1.4)where the superscripts of the metric functions denote the order in the asymptoticexpansion.Since our metric ansatz is not of the Fefferman-Graham form, we need to per-form a change of coordinate (in the x,r plane, for which we used the conformalansatz) to put the metric is such a form. The details of the transformation arestraightforward and the process results in the following shifts in the asymptoticmetric quantities:∆g(3)i j =23g(x), (B.1.5)for every i, j, where g(x) is the leading asymptotic correction to the metric compo-nent grr. That is, at large r that metric component becomesgrr(r,x)→ 12r2 +g(x)r5. (B.1.6)Finally, since the metric becomes diagonal asymptotically, the non-vanishingtime components of the energy-momentum tensor Ttt and Tyt have a simple inter-pretation as energy and momentum density, respectively. The conserved chargesare given by integrating those densities over a spatial slice.B.1.2 Explicit expressions for the chargesHomogeneous solutionFor reference, in this subsection we give the explicit expressions for the homoge-neous RN solution in our conventions. The radius of the horizon is given in termsof the temperature byr0 =16(2piT +√3µ2+4pi2T 2). (B.1.7)182The mass, entropy and charge of the RN solution of fixed length L areMRN =(4r30 +µ2r0)L, (B.1.8)SRN = 4pir20L, (B.1.9)NRN = 2r0µL. (B.1.10)The corresponding densities in the infinite system are given by dividing through byL.Inhomogeneous solutionHere we list explicit expressions for the thermodynamic quantities in our systemin terms of our solution ansatz. Conserved charges are given by integrating overthe inhomogeneous direction. We define f (3) = −(4r30 + µ2r0)/4, the 1/r3 termfrom the function f (r) (equation (4.2.3)), and X (3)(x), for X = {R,S,T}, as thecoefficient of the 1/r3 term of the corresponding metric function. The energy-momentum tensor yields the mass1M =∫ L0〈T tt(x˜)〉dx˜ = 4∫ L0ξ (x)2(− f (3)+5S(3)(x)+3T (3)(x))dx, (B.1.11)the tension in the x directionτx =−∫ L0〈T xx(x˜)〉dx˜ = 2∫ L0ξ (x)2( f (3)+6R(3)(x)+4S(3)(x)+6T (3)(x))dx,(B.1.12)and the pressure in the y directionPy =∫ L0〈T yy(x˜)〉dx˜ =−2∫ L0ξ (x)2( f (3)+6R(3)(x)+10S(3)(x))dx. (B.1.13)Now, expanding the equations of motion at the asymptotic boundary, we get therelation R(3)(x)+2S(3)(x)+T (3)(x) = 0. Using this, we see that 〈T µν(z)〉 is trace-less, as necessary. Conservation of the energy momentum tensor requires ∂xτx = 0.This is related to the constraint equation (4.2.18) and we explain our strategy to1See appendix B.2.3 for details about the numerical process, including the definitions of the x˜coordinate and ξ (x). The functions {R,S,T} are defined on the UV grid; they are analogous to{A,B,C} in the original ansatz.183ensure it is satisfied in appendix B.2.3.The coefficient of the 1/r falloff of the gauge field gives the chargeN =−2∫ L0A(1)t (x). (B.1.14)At the horizon, we read the (constant) temperature asT =18pir0(12r20−µ2)e−(B−A)|r=r0 (B.1.15)and the entropy is proportional to the area of the event horizon, given byS = 4pir20∫ L/40e(B(r0,x)+C(r0,x))dx. (B.1.16)B.1.3 Consistency of the first lawsHere, we discuss the first laws for both the finite length stripe and the stripe on theinfinite domain.Finite systemIn our system, as described above, we have unequal bulk stresses τx2 and Py. Then,if we have a rectangle of side lengths (L,Ly), the work done by the expansion orcompression of this region will differ depending on which direction the stress is in.The usual −PdV term in the first law is replaced and we havedMˆ = T dSˆ+µdNˆ+ τxLydL−PyLdLy, (B.1.17)where the hatted variables represent thermodynamic quantities integrated over theentire system. Defining densities (in the trivial y-direction) byM =MˆLy, S =SˆLy, N =NˆLy, (B.1.18)2We define τx =−Px, where Px is the pressure in the x direction. For our solutions, τx > 0.184we can write the first law asdM = T dS+µdN+ τxdL+dLyLy(−M+T S+µN−PyL). (B.1.19)Tracelessness of the energy-momentum tensor implies M = L(Px+Py), so that theterm proportional to dLy can be rewritten as the conformal identity (4.4.3), whichdisappears for a conformal system described by the first law (4.4.1). Therefore, thefirst law (4.4.1) and the conformal identity (4.4.3) are consistent.Infinite systemFor the infinite system, we define densities in both the x and y directions as equation(4.5.1). Under the scaling symmetry (4.2.19), these scale asm→ λ 3m, s→ λ 2s, n→ λ 2n. (B.1.20)Using the first law (4.5.2), we derive the conformal identity (4.5.3). Again, wecan see this from the first law for the system with integrated charges. Plugging thedensities m,s,n into the first law of the finite length system (4.4.1), we arrive atdm = T ds+µdn+dLL(−m+T s+µs+ τx). (B.1.21)Using the conformal identity of the finite length system (4.4.3), we see that theterm proportional to dL is just the conformal identity for the infinite system, whichis satisfied for a system described by (4.5.2).185B.2 Further Details about the NumericsB.2.1 The linearized analysisFollowing [34], we look for static normalizable modes around the RN background.We consider the fluctuation3δgty = λ((r− r0)rw(r)sin(kx)),δAy = λ (a(r)sin(kx)),δψ = λ (φ(r)cos(kx)), (B.2.1)where λ is a small parameter in which we can expand the equations. Putting thisansatz into (B.2.6) - (B.2.12) and expanding to linear order in λ , we arrive at thelinearized systemw′′(r)− r0a′(r)r3(r− r0) +(4r−2r0)w′(r)r(r− r0) +w(r)(2r0(4r3+4r2r0+4rr02− r0)− k2r2)r2 (4r4− r (4r03+ r0)+ r02) = 0,a′′(r)+(8r4+ r(4r03+ r0)−2r02)a′(r)r (4r4− r (4r03+ r0)+ r02) −k2a(r)4r4− r (4r03+ r0)+ r02 +c1kr0φ(r)√3(4r4− r (4r03+ r0)+ r02)−4rr0w′(r)4r3+4r2r0+4rr02− r0 −4r02w(r)4r4− r (4r03+ r0)+ r02 = 0, (B.2.2)φ ′′(r)+c1kr0a(r)2√3r2 (4r4− r (4r03+ r0)+ r02)−φ(r)(k2+2m2r2)4r4− r (4r03+ r0)+ r02 −(−16r3+4r03+ r0)φ ′(r)4r4− r (4r03+ r0)+ r02 = 0.Fixing the scalar field mass as m2 = −4, there are three parameters in these equa-tions: the temperature of the black brane T0 (equivalently the location of the hori-3Regularity at the black hole horizon enforces that δgty(r0) = 0.186zon r0), the wavenumber k, and the strength of the axion coupling c1. In thisanalysis, we will choose c1 and k and then use a shooting method to find the T0 atwhich normalizable modes appear.Due to the linearity of the equations, the scale of our solutions is arbitrary. Weuse this to fix a Dirichlet condition on w at the horizon. Changing coordinates toρ =√r2− r20, and expanding the equations near ρ = 0 gives regularity conditionson the fluctuations at the horizon in terms of Neumann boundary conditions. Ourhorizon boundary conditions are thenw(ρ)|ρ=0 = 1, w′(ρ)|ρ=0 = a′(ρ)|ρ=0 = φ ′(ρ)|ρ=0 = 0, (B.2.3)Namely, that the fields are quadratic in ρ near the horizon. In order to search fornormalizable modes, we set the sources in the field theory to zero by imposingleading order fall-off conditions near the ADS boundary:w(ρ) =w3ρ3+ . . . , a(ρ) =a1ρ+ . . . , φ(ρ) =φ2ρ2+ . . . . (B.2.4)In practice, after fixing c1 and k, we use T0 as a shooting parameter to find thesolution with the correct w fall-off and the corresponding critical temperature Tc.For each c1, we find a range of unstable momenta. By adjusting the strengthof the axion coupling, one can find a large variation in the size of this unstableregion in the (k/µ,T0/µ) plane (see Fig. B.1). The relationship between c1 andthe maximum critical temperature is well fit by T maxc (c1)/µ = 0.025c1− 0.091.The wavenumbers for the dominant critical modes, corresponding to T maxc (c1), forselect c1 are found in Table B.1.c1 T maxc /µ kc/µ Lµ/4 = pi/2kc4.5 0.012 0.75 2.085.5 0.037 0.92 1.718 0.11 1.3 1.2118 0.37 2.85 0.5536 0.80 5.65 0.28Table B.1: The maximum critical temperatures and corresponding criticalwavenumbers for varying c1.187c1=8c1=36c1=4.5c1=180 2 4 6 8 10 120.00.20.40.60.8k ΜT cΜFigure B.1: The critical temperatures at which the RN black brane becomesunstable, for varying axion coupling c1. As the strength of the axioncoupling increases, the size of the unstable region (the area under thecritical temperature curve) also increases.B.2.2 The equations of motionFor completeness, here we present the equations of motion derived from the La-grangian (4.2.1). The Einstein equations in our case are four second order ellipticequations for the metric components and two constraint equations. For the com-pactness of the expressions, we defineOˆU · OˆV = ∂rU∂rV + 14r4 f ∂xU∂xV, Oˆ2U = ∂ 2r U +14r4 f∂ 2x U. (B.2.5)The four elliptic equations, formed from combinations of Gtt−T tt = 0, Gty−T ty = 0,Gyy−T yy = 0, and Grr +Gxx− (T rr +T xx ) = 0, then take the form188Oˆ2A+(OˆA)2+ OˆA · OˆC− e−2A+2C2 f(OˆW )2− e−2A4r2 f(OˆAt)2− 14r2(e−2AW 2f+ e−2C)(OˆAy)2− e−2AW2r2 fOˆAt · OˆAy+(5r+3 f ′2 f)∂rA+(1r+f ′2 f)∂rC+3r2− 3e2Br2 f+e2Bm2ψ24r2 f+3 f ′r f+f ′′2 f= 0, (B.2.6)Oˆ2B+12(Oˆψ)2− e−2A+2C4 f(OˆW )2− OˆA · OˆC− 1r∂rA+(2r+f ′2 f)∂rB−(1r+f ′2 f)∂rC = 0, (B.2.7)Oˆ2C+(OˆC)2+ OˆA · OˆC+ e−2A+2C2 f(OˆW )2+e−2A4r2 f(OˆAt)2+14r2(e−2AW 2f+ e−2C)(OˆAy)2+e−2AW2r2 fOˆAt · OˆAy+1r∂rA+(5r+f ′f)∂rC+3r2− 3e2Br2 f+e2Bm2ψ24r2 f+f ′r f= 0, (B.2.8)andOˆ2W − OˆA · OˆW +3OˆC · OˆW − e−2CWr2(OˆAy)2−e−2Cr2OˆAt · OˆAy+ 4r ∂rW = 0. (B.2.9)The matter field equations areOˆ2ψ+ OˆA · Oˆψ+ OˆC · Oˆψ+ c1e−A−C8√3r4 f(∂rAt∂xAy−∂xAt∂rAy)+(4r+f ′f)∂rψ− e2Bm2ψ2r2 f= 0, (B.2.10)189Oˆ2At − OˆA · OˆAt + OˆC · OˆAt + e−2A+2CWfOˆW · OˆAt + OˆW · OˆAy+2WOˆC · OˆAy−2WOˆA · OˆAy+ e−2A+2CW 2fOˆW · OˆAy+c14√3r2(eA−C− e−A+CW 2f)(∂rψ∂xAy−∂xψ∂rAy)−c1e−A+CW4√3r2 f(∂rψ∂xAt −∂xψ∂rAt)+ 2r ∂rAt −W f ′f∂rAy = 0, (B.2.11)andOˆ2Ay+ OˆA · OˆAy− OˆC · OˆAy− e−2A+2CWfOˆW · OˆAy− e−2A+2CfOˆW · OˆAt+c1e−A+C4√3r2 f(∂rψ∂xAt −∂xψ∂rAt)+ c1e−A+CW4√3r2 f(∂rψ∂xAy−∂xψ∂rAy)+(2r+f ′f)∂rAy = 0. (B.2.12)Finally, the constraint equations are−2e−2B f r2(∂x∂rA+∂x∂rC)+2e−2B f r2∂rA(∂xB−∂xA)+2e−2B f r2 (∂xA+∂xC)∂rB+2e−2B f r2 (∂xB−∂xC)∂rC− e−2Br2 f ′∂xA+e−2B(r2 f ′+4 f r)∂xB+ e−2(A+B) (∂xAt +W∂xAy)(∂rAt +W∂rAy)+r2e−2(A+B−C)∂xW∂rW − f e−2(B+C)∂xAy∂rAy−2e−2B f r2∂xψ∂rψ = 0(B.2.13)190and∂ 2r A+∂2r C−14 f r4(∂ 2x A+∂2x C)+(1− 14 f r4)(∂rA)2+(1− 14 f r4)(∂rC)2+12 f r4(∂xA+∂xC)∂xB−2(∂rA+∂rC)∂rB+(3 f ′2 f+2r)∂rA−(f ′f+4r)∂rB+(f ′2 f+2r)∂rC+e−2A8 f 2r6(∂xAt +W∂xAy)2− e−2A2 f r2(∂rAt +W∂rAy)2−e−2(A−C)2 f((∂rW )2− 14 f r4 (∂xW )2)+e−2C2r2((∂rAy)2− 14 f r4 (∂xAy)2)+(∂rψ)2− 14 f r4 (∂xψ)2+f ′′2 f+2 f ′f r= 0. (B.2.14)B.2.3 ConstraintsThe constraint equations, Grx−T rx = 0 and Grr−Gxx− (T rr −T xx ) = 0, are the non-trivial Einstein equations that are not part of the system of second-order ellipticequations that we numerically solve. As discussed in §4.2, the weighted constraintscan be shown to solve Laplace equations on the domain. If we satisfy one ofthe constraints on all boundaries and the other at one point, they will be satisfiedeverywhere. At the black hole horizon, we choose to impose r2√f√−g(Grr −Gxx − (T rr − T xx )) = 0 at the point (ρ,x) = (0,0) and√−g(Grx − T rx ) = 0 acrossthe horizon. Since we use periodic boundary conditions in the inhomogeneousdirection, the boundaries at x = 0 and x = xmax are trivial if√−g(Grx− T rx ) = 0at the horizon and the conformal boundary. Then, we are left with the task ofsatisfying√−g(Grx−T rx ) = 0 at the boundary.In §4.2, we found the asymptotic expansion of this constraint asGrx−T rx ∝3∂xA(3)(x)+2∂xB(3)(x)+3∂xC(3)(x)r2+O(r−3), (B.2.15)where A(3)(x),B(3)(x) and C(3)(x) come from solving the elliptic equations. It ap-pears that, within our problem, we do not have the ability to make the weightedconstraint disappear. The key lies in an unfixed gauge symmetry in our original191metric that is related to conformal transformations of the (r,x) plane.4 Essen-tially, within our metric ansatz, we have the freedom to transform to any plane(r′,x′) that is conformally related to (r,x). Demanding that the weighted constraint√−g(Grx−T rx ) vanishes at the conformal boundary uniquely identifies the correctcoordinates (r˜, x˜).Our procedure is to split the domain at some intermediate radial value ρint . Onthe IR portion of the grid, 0 < ρ < ρint , the equations are as above. On the UVportion of the grid, ρint < ρ < ρcut , we use the coordinate freedom to select thecorrect asymptotic radial coordinate. We can write the metric in the UV asds2 =−2r˜2 f˜ (r˜, x˜)e2Rdt2+ e2S(dr˜22r˜2 f˜ (r˜, x˜)+2r˜2dx˜2)+2r˜2e2T (dy−Udt)2,(B.2.16)where f˜ (r˜, x˜) ≡ f (r(r˜, x˜)). Under a transformation in the (r˜, x˜) plane such that r˜and x˜ satisfy Cauchy-Riemann-like relations∂ r˜(r,x)∂ r=r˜(r,x)2r2∂ x˜(r,x)∂x,∂ x˜(r,x)∂ r=− 14r2r˜(r,x)2 f (r)∂ r˜(r,x)∂x, (B.2.17)the metric becomesds2 =−2r˜(r,x)2 f (r)e2Rdt2+ e2S|∇r˜(r,x)|2(dr22r2 f (r)+2r2dx2)+2r˜(r,x)2e2T (dy−Udt)2(B.2.18)with|∇r˜(r,x)|2 = r2r˜(r,x)2(∂ r˜(r,x)∂ r)2+14r2r˜(r,x)2 f (r)(∂ r˜(r,x)∂x)2. (B.2.19)We now have an extra function r˜(r,x) in our system which we may use to satisfythe constraint and fix the residual gauge freedom, as we will now see. The Cauchy-Riemann-like conditions give the Laplace-like equation∂∂ r(r2r˜(r,x)2∂ r˜(r,x)∂ r)+∂∂x(14r2r˜(r,x)2 f (r)∂ r˜(r,x)∂x)= 0. (B.2.20)4See [3] for a discussion of the same issue in a different context.192We can solve this asymptotically, findingr˜(r,x) = ξ (x)r+2ξ ′(x)2−ξ (x)ξ ′′(x)24ξ (x)r+ . . . , (B.2.21)where ξ (x) is an arbitrary function that encodes the coordinate freedom we have.Expanding the constraint asymptotically, we haveGrx−T rx ∝1r2(2(3∂xR(3)(x)+2∂xS(3)(x)+3∂xT (3)(x))ξ (x)+ 3( f (3)+2R(3)(x)−4S(3)(x)+2T (3)(x))ξ ′(x))+O(r−3),(B.2.22)where X = X (3)(x)/r3+ . . . asymptotically, for X = {R,S,T}. Demanding that theconstraint (B.2.22) vanishes at the leading order yields a differential equation wecan solve for ξ (x), giving us a boundary condition for the function r˜(r,x), such thatthe weighted constraint will disappear at the conformal boundary. However, wehave found that the code is unstable if we directly use this solution for ξ (x). Insteadof directly integrating the constraint, we use the freedom in ξ (x) to fix the tensionτx to be constant. This enforces the same effect on the tension as if we had used theexplicit solution for ξ (x) but is much more stable numerically. Below, we checkthat the constraints are suitably satisfied even though our boundary conditions donot exactly fix them. To this end, we setξ (x) =K( f (3)+6R(3)(x)+4S(3)(x)+6T (3)(x))1/3. (B.2.23)Expanding the equations asymptotically gives the expression R(3)(x)+ 2S(3)(x)+T (3)(x) = 0; if this is satisfied on our solutions our definition of ξ (x) coincides withthat found by integrating the constraint (B.2.22).The constant K appearing in ξ (x) sets the scale of the boundary theory. Weuse it to fix the length of the inhomogeneous direction in the field theory to beLµ/4. The correct coordinate in the inhomogeneous direction of the field theoryis x˜. From the Cauchy-Riemann conditions, we can find the large r expansion of193x˜(r,x) asx˜(r,x) =∫ x0dx′ξ (x′)+ξ ′(x)8ξ (x)2r2+ . . . . (B.2.24)Integrating to find the proper length of one cycle in the boundary, we solve for Kat leading order in r to findK =4L∫ L/40( f (3)+6R(3)(x)+4S(3)(x)+6T (3)(x))1/3dx′. (B.2.25)When integrating the charges over the inhomogeneous direction in the field theory,one must remember to integrate over the correct coordinate, dx˜ = dx/ξ (x).Our corrected numerical procedure is as follows. On the IR grid, we solve theelliptic equations (B.2.6) - (B.2.12) for the metric functions A,B,C and W . Onthe UV grid, we solve the equivalent elliptic equations from the metric (B.2.18)in the variables R,S,T and U plus equation (B.2.20) for the new field r˜(r,x). Atthe horizon, we enforce the boundary conditions discussed in §4.2. At the inter-face ρ = ρint , we impose matching conditions on the four metric functions andthat r˜(ρint ,x) = r(ρint). Asymptotically, R,S,T and U all fall off as 1/r˜3. To setboundary conditions on r˜, we notice that∂r r˜(r,x)+r˜(r,x)r= 2ξ (x)+O(1r3). (B.2.26)We truncate this expression at O(r−2) and finite difference to find an update proce-dure for r˜(ρcut ,x). This boundary condition is updated iteratively as the functionsR,S,T are updated in our solving procedure such that once we find a solution withsmall residuals we can be sure that the tension is constant and the constraint issatisfied.B.2.4 Generating the action density plotTo generate the relative action density plot, Fig. 4.17, we find the solutions ona grid of lengths L and temperatures T0, as shown in Fig. B.2. By interpolatingthese solutions on the domain, we can map the thermodynamic quantities acrossthe unstable region and determine the approximate line of minimum free energy,or the dominant solution in the infinite size system.1941.0 1.2 1.4 1.6 1.8 2.0 2.20.000.020.040.060.080.10L̐4TΜFigure B.2: The data underlying Fig. 4.17. The points represent solutionswe computed. These were interpolated to find the free energy densityover the domain. The solid blue line is the edge of the unstable regionand the thick red line is the approximate line of minimum free energydensity.B.2.5 Convergence and independence of numerical parametersPerformance of the method and convergence of physical dataAs discussed above, to solve the equations numerically, we use a second orderfinite differencing approximation (FDA) before using a point-wise Gauss-Seidel,SOR relaxation method on the resulting algebraic equations. The method, includingthe UV procedure described above, performs well for this system.The UV procedure is unstable for a generic initial guess, resulting in a diver-gent norm. To find a solution from a generic initial guess, we can run the relaxationwithout the UV procedure until the norm is small enough that the result approxi-mates the true solution, before activating the UV procedure to find the true solution.Once we have these first solutions, by using these as an initial guess for solutionsnearby in parameter space and by interpolating to a finer grid, we can generate fur-ther solutions by relaxing with the UV procedure. In Fig. B.3, we plot the L2 normof the total residual during the relaxation of the c1 = 8 solution at T0 = 0.04 and1950 500 000 1.0´ 106 1.5´ 10610-710-50.0010.110iterationsÈresidual L2Figure B.3: The behavior of the L2 norm of the residual during the relaxationiterations for c1 = 8, T0 = 0.04 and Lµ/4 = 0.75. From top to bottom(at the left of the plot) the grid spacing is dρ,dx = 0.04,0.02,0.01.The UV procedure is unstable unless the solution is close enough tocorrect solution. For grid spacing dρ,dx = 0.04, the UV procedure wasactivated after 3× 105 iterations while for the others, the initial guesswas taken to be a solution with slightly different parameters such thatthe UV procedure could be used immediately.Lµ/4 = 0.75 for the grid spacings dρ,dx = 0.04,0.02,0.01, showing the expectedexponential behavior of the Gauss-Seidel relaxation. The physical data extractedfrom our solutions is consistent with the expected second order convergence of ourFDA scheme, see Fig. B.4.Asymptotic versus first law massA useful check of the numerics is to compare the mass of the system read offfrom the asymptotics of the metric, equation (B.1.11), to that computed by inte-grating the first law, equation (4.4.1). Since the temperature and entropy are readoff from the horizon, comparing these two methods of finding the mass provides anon-trivial global consistency check on our results. We verify that the differencebetween the asymptotic mass and the first law mass remains smaller than 0.5%across our set of trials, indicating consistency of our results.1960.0 0.2 0.4 0.6 0.80.01250.01300.01350.01400.01450.01500.01550.0160TTc<OΨ>Μ2Figure B.4: The value of the scalar field condensate for varying grid sizesfor c1 = 8 and Lµ/4 = 0.75. From top to bottom, the grid spacing isdρ,dx = 0.01,0.02,0.04. The results are consistent with second orderscaling as expected from our numerical approach.A related check of the numerics is the conformal identity or the Smarr-likerelation, 2M = T S+µN−τxL, derived above from the first law for the finite lengthsystem. To evaluate how well our solutions satisfy this equation, we examine theratio2M f all−o f f −T S−µN+ τxLmax(M f all−o f f ,T S,µN,τxL), (B.2.27)since the largest term in the expression sets a scale for the cancellation we expect.This ratio is very small for our solutions near the critical temperature. As we lowerthe temperature, this ratio increases, but stays small. The precise value depends onthe parameters of the solution, but is not larger than order 1%. Moreover, this ratiodecreases as we move the position of the finite cutoff of the conformal boundary toa larger radius.Finite ρcut boundary checkFor the c1 = 8 trials reported in the paper, we use ρcut = 12 as our conformalboundary. In Table B.2 we present results for varying ρcut , showing that our choiceis large enough such that the physical results are insensitive to the cutoff. Although197the physical results presented in the table appear very stable, at small ρcut , theresults for the mass and charge depend significantly on the fitting procedure forthe asymptotic metric functions and gauge field. By running our simulations atρcut = 12, we are both well within the the region where the solutions do not changewith the conformal boundary and within a region where our fitting procedure to theasymptotics behaves well.ρcut S M N1 0.758504 0.305774 0.5274062 0.767913 0.342327 0.4905243 0.768211 0.341928 0.4905934 0.768285 0.342043 0.4905835 0.768311 0.342136 0.4905776 0.768322 0.34221 0.4905747 0.768328 0.342277 0.4905728 0.768332 0.342324 0.490579 0.768334 0.342367 0.49056910 0.768335 0.342402 0.49056811 0.768336 0.342434 0.49056812 0.768336 0.342459 0.490567Table B.2: Behavior of physical quantities with the cutoff for c1 = 8 andLµ/4 = 0.75 and for fixed grid resolution dρ,dx∼ 0.02. The entropy Sis read off at the horizon, while the mass M and the charge N are readoff at the conformal boundary. Both the entropy and the charge are veryrobust against the location of the conformal boundary. The mass takesslightly longer to settle down, but is well within the convergent range forρcut = 12.Behavior of the constraintsOne of the most important checks for our numerical solution is the behavior of theconstraints. For numerical homogeneous solutions found with our method, the L2norm of the constraints is very small, on the order of 10−4. For the inhomogeneoussolutions, the constraints are small near the critical temperature, but grow and sat-urate as we lower to the temperature, to have a maximum L2 norm on the orderof 10−2: see Fig. B.5. Since our boundary conditions explicitly fix the weighted198r2√f√−g(Grr−Gxx− (T rr −T xx ))√−g(Grx−T rx )Figure B.5: The weighted constraints for c1 = 8 and Lµ/4 = 1.21. The topplots are near the critical point, T/Tc = 0.97, while the bottom plots areat small temperature, T/Tc = 0.00016. By our boundary conditions,the constraints disappear at the horizon. They approach a finite value asthey approach the asymptotic boundary.constraints on the horizon, they disappear there. The weighted constraints thenincrease towards the conformal boundary, approaching a modulated profile of con-stant amplitude. The amplitude near the conformal boundary controls the overallL2 norm of the constraints.The constraint violation improves marginally with step size and with movingthe interface closer to the horizon, but does not improve as we take the conformalboundary to a larger radius. To check that the constraints are well satisfied on oursolution, we compare them to the sum of the absolute value of the terms that makeup the constraints. That is, if the constraints are given by ∑i hi, we compare this199to ∑i |hi|. This procedure gives us an idea of the scale of the cancellation amongthe individual terms hi. We find that the sum ∑i |hi| diverges approximately as r4towards the asymptotic boundary, such that the approach of the constraint violationto a constant is a good indicator that the constraints are satisfied on the solution.In Table B.3, we compare the L2 norm of these two sums on the entire domain,showing that the constraint violation for the inhomogeneous solutions is generallyabout four orders of magnitude less than the scale set by ∑i |hi|. Interestingly, therelative constraint improves marginally as we go to lower temperatures.Parameters T0 L2(∑i hi)/L2(∑i |hi|)c1 = 8,Lµ/4 = 2.00 (RN solution) 0.105 9.12 ·10−7c1 = 8,Lµ/4 = 1.21 (striped solution) 0.075 2.02 ·10−40.05 1.84 ·10−40.025 1.58 ·10−40.005 1.37 ·10−40.001 1.32 ·10−4Table B.3: Comparison of the constraint violation, measured by theschematic constraint equation ∑i hi, to the scale set by the individualterms, ∑i |hi|, for grid size dρ,dx ∼ 0.01. We take the L2 norm of themeasures on the entire domain. The c1 = 8,Lµ/4 = 2.00 solution is ahomogeneous RN solution found numerically with our code, for whichthe constraints are very well satisfied. The constraints for the striped so-lutions are satisfied compared to the scale set by ∑i |hi| by four orders ofmagnitude and the relative constraint improves marginally as we lowerthe temperature.The asymptotic equation of motionExpanding the equations of motion asymptotically gives the relationR(3)(x)+2S(3)(x)+T (3)(x) = 0, (B.2.28)which can be used to give another check of the numerics. As explained in B.1.2,this condition implies the tracelessness of the energy-momentum tensor. For theinhomogeneous solutions near the critical temperature we find that this expressionis on the order of the individual metric functions X (3), where X = {R,S,T}, but200generally decreases as we lower the temperature. As well, we find that homoge-neous solutions found using our numerical techniques satisfy (B.2.28) well. Thereseems to be an unidentified systematic error here that may deserve further attentionin the future. Possible problems may occur in the implementation of the UV pro-cedure or in our procedure to read off the coefficients of the falloffs of the metricfunctions. However, our physical results are robust under changes to the boundaryconditions, so that we are confident in our results despite this possible systematic.In particular, the physical quantities extracted from the horizon are independentof the different boundary constraint fixing schemes we implemented. Therefore,we advocate using the mass derived from the integrated first law, which uses noasymptotic metric functions.201Appendix CAppendix: Fermi Liquids fromD-branesC.1 Charges and Stress Tensor ComponentsThe sources in the equations of motion correspond to expectation values of fermionbilinears, with factors of k and ω absorbed into the definition for later numericalconvenience. In combining these Grassmann quantities into bosonic fields it wasimportant to specify a basis, which we choose to be satisfy gg∗∧gg = 1. Here ggis taken to be an arbitrary spinor. The definitions of the bilinears are thenQ11 =∫ k f0kdkpi〈 f ∗1 f1〉, Q22 =∫ k f0kdkpi〈 f ∗2 f2〉, Q12 =∫ k f0kdkpi〈 f ∗2 f1〉,Q21 = Q12, P11 =∫ k f0kdkpik〈 f ∗1 f1〉, P22 =∫ k f0kdkpik〈 f ∗2 f2〉,L11 =∫ k f0kdkpiω(k)〈 f ∗1 f1〉, L22 =∫ k f0kdkpiω(k)〈 f ∗2 f2〉 (C.1.1)We identify the combinations of Q = (Q11+Q22), Tr = (Q12+Q21), and TM =(L11+L22+P11−P22) as the charge, and radial and Minkowski “stress-energy”components, respectively. The current is the source term for the gauge field, whilein analogy to the gravitational case we refer to the sources for the embedding func-202tion as “stress-energy”. Using the form1 of the fermion contribution to the stressenergyT MNf ermi = βi2(ψ¯Γ(MDN)ψ−D (Mψ¯ΓN)ψ) (C.1.2)and the rescaling of the fermions given in Equation 5.2.7, the “radial” and “Minkowski”stresses are seen to result from this expression by taking the fermion derivativeswith respect to the radial and boundary directions, respectively.C.2 NumericsFinding the solutions presented in this paper required the use of Mathematica andMatlab for symbolic and numerical work, respectively. This appendix outlinessome of the key tools used, pitfalls encountered and presents some convergencetests.C.2.1 Numerical techniquesThe derivation of the equations was done using Mathematica, utilizing the packageMathTensor [23]. The Grassmann package [96] was useful for consistent manipu-lation of anti-commuting fields. An efficient way to export the equations to Matlabis using the freely available ToMatlab package [66]. In Matlab the equations weresolved using the procedure described in Section 5.3. Here there are several pointsworthy of note:• The general setup of the spectral code follows the useful pedagogical ref-erence [126]. Once the equations were discretized, the fermionic eigen-value problem was solved using Matlab’s “eigs” function. The source termswere then calculated via appropriate numerical integration. In calculating thesource terms it was important to normalize the eigenvalues via∫ pi/20 dψ¯ψ =1. This is necessary to ensure a consistent definition of the fermion energy1We note that without the symmeterization thus is just the usual fermion stress energy tensor, ascalculated via Noether’s theorem. The additional symmeterization makes the stress energy mani-festly symmetric and is most simply obtained via appropriate variation of the Dirac action in termsof the metric as in [130]. In our case we must pullback the bulk form of this tensor onto the braneworldvolume.203ω(k) and therefore consistent charges, J, Tr, TM. It can be easily checkedthat with this normalization∫T00 dr = w(k) = E in the limit of a flat in-duced metric and zero gauge field.• The discretized bosonic system was solved via a generalization of the New-ton’s method (see [17] for an a good reference on this and other spectralmethods). It was also necessary to add a line search algorithm to this solver,as outlined in [109], to improve convergence and stability properties.• It is well known that spectral methods generally respond poorly to non-analytic behaviour. Thankfully in our case the simple asymptotic behaviourof all fields ensures a good convergence of spectral methods.• It was found that the bosonic and fermionic equations were regular at theembedding cap-off and at the conformal boundary. Therefore it was notnecessary to impose cutoffs to regulate the problem.C.2.2 Convergence checksWe now consider the accuracy of our numerical solutions. W present a seriesof plots which illustrate that our solutions converge to solutions of the integro-differential equations in the large N limit (here N is the order of the Chebyshevapproximation not the size of the adjoint gauge group). It was particularly impor-tant to verify that our solutions for the embedding function are numerically robust.The key convergence criteria were:• The stability and accuracy of the eigenvalue routines – this was the easiestpart to check as these were based on built-in Matlab functions. The defaultaccuracy in using the “eigs” routine is machine precision. Therefore this wasnot considered a significant source of error.• It was necessary to ensure that the both the L 2 norm of the residues ofthe bosonic system is and the maximum change in the bosonic fields after afermion/boson iteration loop were both driven to numbers of O(10−12).20460 80 100 120 140 160 180 200 22000.20.40.60.811.21.41.61.8x 10−8 G convergence with NNmax |∆ G|60 80 100 120 140 160 180 200 22000.511.522.533.54x 10−9 u convergence with NNmax |∆ u|60 80 100 120 140 160 180 200 22000.10.20.30.40.50.60.70.80.91x 10−3 J convergence with NNmax |∆ J|60 80 100 120 140 160 180 200 22000.511.522.53x 10−3TM convergence with NNmax |∆ T M|60 80 100 120 140 160 180 200 22000.20.40.60.811.2x 10−3Tr convergence with NNmax |∆ T r|Figure C.1: The convergence with N for the embedding function, gauge field,and charges. The maximum difference between solutions between suc-cessive values of N is computed, and can be seen to converge to zero asN is taken large. The convergence tests plotted were run at parametervalues of µ =−15.7154, m0 = 1, mpsi = 10, ε = 0.01, β =−0.01.205• The solution of the discretized equations converged as N was increased there-fore indicating that we may be tending to a solution of the original ODEequations- see figure Figure C.1.• It was necessary to check that integration procedures used in computing thecharges were sufficiently accurate, and the change in the solutions with in-creasing tolerance tended to zero.206Appendix DAppendix: Spatial Modulationand Conductivities in EffectiveHolographic TheoriesIn this appendix we provide further information on our numerical methods and ex-hibit some checks on the results presented in this paper. We first briefly describe ourapproach to solving the background and perturbation equations. We then presentour convergence results and other checks of the numerics.D.1 Numerical Procedure and Implementation DetailsThe derivation of the equations, boundary conditions and gauge conditions weredone in Mathematica. These expressions, in appropriately discretized form, werethen exported to Matlab which served as the principal platform for equation solvingand post-processing work. Significant portions of the code were transferred to C++code, which utilized the Armadillo and BLAS linear algebra libraries.The non-linear nature of the background equations of motion requires an it-erative process for the solution. This was accomplished using a combination ofNewton method with line search and the quasi-Newton, Broyden method algo-rithms. The Broyden method was found to be the most efficient approach, as it didnot necessitate the computationally expensive process of assembling the Jacobian207matrix. As the results presented in this paper rely on knowledge of the tempera-ture dependence of the conductivity at each point in the υ ,k parameter space, itwas necessary to numerically solve the PDEs many times. As such the utilizationof optimized code and efficient solver algorithms was important in maintaining amanageable computational load. An example of a useful solver strategy for thebackground was:• First attempt to use the Broyden method. If the initial guess is sufficientlyclose to the sought after solution, convergence should be reached rapidly.• If the Broyden method fails to converge the solver switches to Newton methodaugmented with a three-point safeguarded parabolic line search. Once thenorm of the residual has reduced below a safe tolerance, the Broyden methodcan once again be utilized to quickly bring about convergence to the requiredaccuracy.• Once an initial solution is obtained suitably sized (adiabatic) variations in pa-rameters allow for the mapping of parameter space, without having to resortto the Newton method.The solution of the perturbation equations is in some sense easier as the linearnature of the equations means that they can be inverted in one step without needfor iteration. An additional complication is introduced by the fact that knowledgeof the temperature dependence of the AC conductivity requires us to scan over bothtemperature and frequency at each value of the parameters υ ,k. As the quality ofthe numerical convergence is not uniform with temperature or frequency, we mustbe careful to establish which regions of the (T,ω,υ ,k) space we can reliably probeand bear this in mind when analyzing our results. We also must be careful to ensurethat we have sufficiently resolved the background solution relative to the desiredresolution for the perturbation solution.D.2 Convergence TestsWe now present the following convergence results for the background and pertur-bation solutions:208• The background solution:– Convergence of the solutions to equations Equation 6.2.2 as a functionof the grid size N.– Convergence of the DeTurck gauge condition to zero as a function ofN.• The linear perturbation solutions:– Convergence as function of the grid size Np for the linearized equa-tions.– Convergence of the deDonder and Lorentz gauge conditions Equa-tion 6.2.11 to zero as a function of Np.– Convergence of the auxiliary (unimposed) horizon boundary condi-tions, as described below equation Equation 6.2.13, to zero as functionof Np.In Figure D.1 and Figure D.2, we consider the convergence of the solutions asa function of N, for various low temperatures and υ . We consider the convergenceproperties of the solutions as a function of both the transverse grid size, Nx, andradial grid size, Nz, separately. This may be done by fixing one of the grid sizesand running convergence tests on the other. We display the results below for theconvergence of the log of the norm of the difference in solutions for three separate,low temperatures. In the first set we fix Nx = 45 and vary Nz while in the second wedo the reverse. From these tests we may draw the (perhaps expected) conclusionthat convergence of the solutions depends more strongly on the radial grid.We note that the asymptotic expansion of the fields near the conformal bound-ary contain logarithmic terms at high orders of the expansion. Therefore the expo-nential convergence of the spectral methods we are using is expected to fail for fineenough grids. However, we find that for the range of parameters we consider here,this is not an important issue. Similarly, at sufficiently low temperatures we expectthat finite differencing approximation near the horizon is more suitable. It may bethat utilizing such methods together with domain decomposition approaches willallow us to reach still lower temperatures in future work.20930 35 40 45 50 55 60 65−10−9−8−7−6−5−4Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.0935 40 45 50 55 60 65−11−10−9−8−7−6−5−4Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.09Figure D.1: Convergence of the log of the norm of the difference of the so-lutions as a function of Nz for temperatures of (0.0016,0.0021) for avariety of υ with Nx = 45. These temperatures correspond to Tµ valuesof approximately (6.6e−4,8.6e−4) and therefore can be considered tobe small on the scale set by the chemical potential. We see that we ob-tain exponential convergence as a function of Nz, and that, while theseplots become noisier for smaller temperatures, the exponential conver-gence reasserts itself as Nz increases.21015 20 25 30 35 40 45 50−20−19−18−17−16−15−14−13−12Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.0915 20 25 30 35 40 45 50−20−19−18−17−16−15−14−13−12Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.09Figure D.2: Convergence of the log of the norm of the difference of the so-lutions as a function of Nx for the same values of υ and temperature asthe previous plot and with with Nz = 45. We note that the convergencedeviates significantly from exponential and that the small scale of they axis indicates that convergence is occurring more slowly than for thecase of increasing Nz.211In Figure D.3 we perform a similar series of tests for the convergence of thenorm of the gauge condition towards zero. We again find that better convergencebehaviour is achieved by increasing Nz in preference to Nx. We also note thatdifferent scaling regimes may exist in the convergence of the norm of the gaugecondition as a function of the resolution, cf., Figure D.4.From this series of experiments we conclude that the best resolution for a fixednumber of grid points is obtained when more points are allocated to the radial gridnumber, Nz. As a verification of this approach in Figure D.5 we plot the log of thenorm of the residues as above with Nz and Nx both increasing but with Nz =Nx+20.We now consider convergence results for the perturbation equations. Firstly itwas noted that asymmetric grids are less useful in this case and that the best resultswere obtained when Nx and Nz were increased in tandem. In addition it was foundthat for smooth convergence to be obtained the background should be available ata higher resolution then the perturbation resolution. We are now interested in theconvergence as a function of υ , T and ω .For our purposes there are two w regimes where we must examine the conver-gence of the AC conductivity. These are w 1, which is relevant for comparisonto the DC conductivity, and 5∼<w<∼ 10 which is relevant for examining poten-tial IR scaling regimes. In Figure D.6 and Figure D.7 we display some examplesof the convergence of the norm of the difference of the perturbative solutions, andthe norm of the gauge and auxiliary conditions as a function of N for w = 0.1. InFigure D.8 and Figure D.9 we plot the analogous results for the real and imaginaryparts of the conductivities themselves. We note that as Tµ is lowered and υ increasedthe convergence of the norm of the gauge and auxiliary conditions becomes morestrained. We have however checked that for the data displayed in Figure 6.1 bothare on the order of 10−5 when the DC data points are read off.The situation is more complicated in the opposite limit which we examine inFigure D.10 and Figure D.11, and Figure D.12 and Figure D.13 where w = 7.We note that while convergence is maintained, the rate of convergence for theauxiliary horizon constraints becomes problematic as the temperature is lowered.It is instructive to examine the form of these auxiliary constraints correspondingto the results displayed Figure 6.11 and Figure 6.12, and Figure 6.13 and Fig-ure 6.14. This is done in in Figure D.14 and Figure D.15, and Figure D.16 and21230 35 40 45 50 55 60 65−8.5−8−7.5−7−6.5−6−5.5−5−4.5−4Convergence of norm of gaugeNLog of norm of gauge  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.0935 40 45 50 55 60 65−9.5−9−8.5−8−7.5−7−6.5−6−5.5−5−4.5Convergence of norm of gaugeNLog of norm of gauge  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.09Figure D.3: Examining the convergence of the log of the norm of the gaugecondition as a function of Nz. For the lower temperature (leftmostgraph) we observe exponential convergence with the now expectednoise at lower values of Nz. We note that for the higher temperature(rightmost graph) larger magnitudes of υ exhibit two distinct scalingregimes with a crossover occurring at approximately Nz = 55. Whilethe convergence is exponential in both cases it is markedly faster in onecase. This may mean that it is necessary to go to higher resolutions ifvery accurate solutions are required in this region of parameter space.This issue however was not encountered in the results presented earlierin the paper.21315 20 25 30 35 40 45 50−6.8−6.6−6.4−6.2−6−5.8−5.6−5.4−5.2−5−4.8Convergence of norm of gaugeNLog of norm of gauge  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.0915 20 25 30 35 40 45 50−7.8−7.6−7.4−7.2−7−6.8−6.6−6.4−6.2−6−5.8Convergence of norm of gaugeNLog of norm of gauge  υ=0.01υ=0.03υ=0.05υ=0.07υ=0.09Figure D.4: For the same values of υ and temperature as Figure D.3. We seethat the convergence of the log of the norm of the gauge condition issignificantly slower then exponential when Nx is increased for a fixedNz.21435 40 45 50 55 60 65−10−9.5−9−8.5−8−7.5−7−6.5−6−5.5−5Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.08υ=0.09υ=0.135 40 45 50 55 60 65−12−11−10−9−8−7−6−5Convergence of norm of residuesNLog of norm of residues  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.08υ=0.09υ=0.1Figure D.5: Convergence of the log of the norm of the residues with both Nzand Nx increasing but with Nx lagging Nz by 20. Again the Tµ values areapproximately (6.6e−4,8.6e−4) moving from left to right. We see goodexponential convergence even for the lower temperature case.21510 15 20 25 30 35 40 45−24−22−20−18−16−14−12−10Norm of the difference at T/µ=0.052109 ω=0.0098 ω/T=0.1NLog of norm of the difference in solutions  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40 45−16−14−12−10−8−6−4−20Norm of the difference at T/µ=0.0011499 ω=0.00028 ω/T=0.1NLog of norm of the difference in solutions  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40 45−11−10−9−8−7−6−5−4Norm of auxiliary conditions at T/µ=0.052109 ω=0.0098 ω/T=0.1NLog of auxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.6: Semi-logarithmic plots of the norm of the difference in solutions,and the norm of horizon auxiliary conditions and of the gauge condi-tions versus N at a low and high temperature for wT = 0.1. We note thatwhile still convergent, the resolution of the auxiliary conditions mustbe monitored closely for small values of Tµ and larger values of υ .21610 15 20 25 30 35 40 45−4−202468Norm of auxiliary conditions at T/µ=0.0011499 ω=0.00028 ω/T=0.1NLog of auxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40 45−16−15−14−13−12−11−10−9−8Norm of gauge conditions at T/µ=0.052109 ω=0.0098 ω/T=0.1NLog of gauge condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40 45−13−12−11−10−9−8−7−6−5−4−3Norm of gauge conditions at T/µ=0.0011499 ω=0.00028 ω/T=0.1NLog of gauge condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.7: A continuation of the sequence of figures begun in Figure D.6.21720 22 24 26 28 30 32 34 36 38 408.688.698.78.718.728.738.748.758.768.77Real conductivity at T/µ=0.052109 ω=0.0098 ω/T=0.1NReal Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.120 22 24 26 28 30 32 34 36 38 4051015202530Real conductivity at T/µ=0.0011499 ω=0.00028 ω/T=0.1NReal Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.8: Plots of the real and imaginary conductivities versus N for thesame choice of parameters as in Figure D.6 and Figure D.7. We notethat, as expected, convergence is slower for larger values of υ . This isparticularly evident for the imaginary part of the conductivity21820 22 24 26 28 30 32 34 36 38 401.1251.131.1351.141.1451.151.1551.161.165Imaginary conductivity at T/µ=0.052109 ω=0.0098 ω/T=0.1NImaginary Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.120 22 24 26 28 30 32 34 36 38 40−100−80−60−40−200204060Imaginary conductivity at T/µ=0.0011499 ω=0.00028 ω/T=0.1NImaginary Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.9: A continuation of the sequence of figures begun in Figure D.8Figure D.17, respectively. We see that the constraint violation worsens as the tem-perature is decreased or υ increased, and that the violation is worst in the regimeof 3∼< w<∼ 5. It has been observed that as one adjusts parameters further intothese regimes numerical artifacts appear in both the imaginary conductivity andthe diagnostic function, F(w). However, given that the conditions are under bettercontrol for w> 5, we believe changed that the results displayed in Figure 6.11 andFigure 6.12, and Figure 6.13 and Figure 6.14 are qualitatively correct. Future workin this direction may involve the use of alternative numerical methods (for example21910 15 20 25 30 35 40−18−17−16−15−14−13−12−11−10Norm of the difference at T/µ=0.027214 ω=0.406 ω/T=7NLog of norm of the difference in solutions  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40−13−12−11−10−9−8−7−6Norm of the difference at T/µ=0.0025647 ω=0.0434 ω/T=7NLog of norm of the difference in solutions  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40−8−7.5−7−6.5−6−5.5−5−4.5−4−3.5Norm of auxiliary conditions at T/µ=0.027214 ω=0.406 ω/T=7NLog of auxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.10: Semi-logarithmic plots of the norm of the difference in solu-tions, and the norm of horizon auxiliary conditions and of the gaugeconditions versus N at a low and high temperature for wT = 7. We notethe unusual dual scaling regimes for the gauge conditions in the highertemperature case. More importantly we note the slow rate of conver-gence of the auxiliary conditions as the temperature is decreased.22010 15 20 25 30 35 40−3−2−10123Norm of auxiliary conditions at T/µ=0.0025647 ω=0.0434 ω/T=7NLog of auxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40−13.5−13−12.5−12−11.5−11−10.5−10−9.5−9−8.5Norm of gauge conditions at T/µ=0.027214 ω=0.406 ω/T=7NLog of gauge condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.110 15 20 25 30 35 40−13−12−11−10−9−8−7−6Norm of gauge conditions at T/µ=0.0025647 ω=0.0434 ω/T=7NLog of gauge condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.11: A continuation of the sequence of figures begun in Figure D.10.22120 22 24 26 28 30 32 34 36 38 400.3820.3840.3860.3880.390.3920.3940.3960.3980.4Real conductivity at T/µ=0.027214 ω=0.406 ω/T=7NReal Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.120 22 24 26 28 30 32 34 36 38 404681012141618Real conductivity at T/µ=0.0025647 ω=0.0434 ω/T=7NReal Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.12: Plots of the real and imaginary conductivities versus N for thesame choice of parameters as in Figure D.10 and Figure D.11. Againwe note that increasing υ makes the convergence more difficult. Inthis case the effect is most notable in the real part of the conductivity.22220 22 24 26 28 30 32 34 36 38 401.3851.391.3951.41.4051.411.4151.421.4251.43Imaginary conductivity at T/µ=0.027214 ω=0.406 ω/T=7NImaginary Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.120 22 24 26 28 30 32 34 36 38 400510152025Imaginary conductivity at T/µ=0.0025647 ω=0.0434 ω/T=7NImaginary Conductivity  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.08ν=0.09ν=0.1Figure D.13: A continuation of the sequence of figures begun in Figure D.12finite difference discretization) to tackle these numerically difficult regimes.D.3 Fit to Drude FormAnother useful test, described in [73], is comparing the low frequency behaviourof the AC conductivity to the Drude form of the conductivity, expected on general2230 1 2 3 4 5 6 7 800.010.020.030.040.050.060.070.080.09Fit of auxiliary condition T/µ=0.019889w/TAuxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 1 2 3 4 5 6 7 800.020.040.060.080.10.12Fit of auxiliary condition T/µ=0.015949w/TAuxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure D.14: The auxiliary horizon constraints corresponding to Figure 6.11and Figure 6.12. We see that while very well satisfied in the w→ 0limit there is a regime of significant constraint violation prior to thereturn of greater accuracy in the mid IR.2240 1 2 3 4 5 6 7 800.020.040.060.080.10.120.140.160.18Fit of auxiliary condition T/µ=0.012167w/TAuxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.10 1 2 3 4 5 6 7 800.050.10.150.20.250.30.35Fit of auxiliary condition T/µ=0.0085301w/TAuxiliary condition norm  ν=0.01ν=0.02ν=0.03ν=0.04ν=0.05ν=0.06ν=0.07ν=0.09ν=0.1Figure D.15: A continuation of the sequence of figures begun in Figure D.14grounds [60]:σ(ω) =K τ1− iω τ (D.3.1)where K is a constant. This test is useful only in the metallic phase or abovethe critical temperature for the onset of the insulating phase. Therefore at mod-erate values of Tµ the AC conductivity is well modelled by the Drude behaviouras demonstrated by the examples in Figure D.18 and Figure D.19.This may be in-2250 1 2 3 4 5 6 7 800.050.10.150.20.250.30.35Fit of auxiliary condition ν=0.1w/TAuxiliary condition norm  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030 1 2 3 4 5 6 7 800.050.10.150.20.250.30.35Fit of auxiliary condition ν=0.09w/TAuxiliary condition norm  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure D.16: The auxiliary horizon constraints corresponding to Figure 6.13and Figure 6.14. Again we observe the non-trivial dependence of theauxiliary constraints on the temperature, υ and w with greater accu-racy being obtained at lower and higher values of w.2260 1 2 3 4 5 6 7 800.010.020.030.040.050.060.07Fit of auxiliary condition ν=0.02w/TAuxiliary condition norm  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.0240030 1 2 3 4 5 6 7 800.0050.010.0150.020.0250.030.035Fit of auxiliary condition ν=0.01w/TAuxiliary condition norm  T/µ=0.0085301T/µ=0.012167T/µ=0.015949T/µ=0.019889T/µ=0.024003Figure D.17: A continuation of the sequence of figures begun in Figure D.16terpreted as an important additional test of the numerics- the quality of the fit tothe Drude form being indicative of good convergence of the perturbation equationsover the range of ω examined.An additional important test of the Drude behaviour is that the coefficient, K,should match with the residue of the zero frequency pole in the imaginary conduc-tivity obtained in the homogeneous case:Im(σhom(ω))→ Kω , as ω → 0 (D.3.2)2270 0.02 0.04 0.06 0.08 0.1 0.12 0.1411.522.533.544.5Drude fit of imaginary conductivity as a function of temperaturewIm(σ)  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.09υ=0.10 0.02 0.04 0.06 0.08 0.1 0.12 0.1423456789Drude fit of real conductivity as a function of temperaturewRe(σ)  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.09υ=0.10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080.511.522.533.544.555.5Drude fit of imaginary conductivity as a function of temperaturewIm(σ)  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.09υ=0.1Figure D.18: The fits of the Drude parameters τ,K to the AC electric conduc-tivities for two moderate values of Tµ , corresponding to those listed inTable D.1, for 0.01 ≥ υ ≤ 0.1. We see that the fits are good, particu-larly for lower values of the ω where the convergence behaviour of theperturbation equations, and associated gauge and auxiliary conditionsis best. In both cases the maximum ωT values plotted areωT ∼ 1.25.2280 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.084567891011Drude fit of real conductivity as a function of temperaturewRe(σ)  υ=0.01υ=0.02υ=0.03υ=0.04υ=0.05υ=0.06υ=0.07υ=0.09υ=0.1Figure D.19: A continuation of the sequence of figures begun in Figure D.18In order to perform this calculation we solved the ODEs which govern the homo-geneous phases of this model at the same point in parameter space as our PDEsEquation 6.2.2 and extracted the residue of the pole. These ODEs may be derivedfrom the (gauged) PDEs via the following substitution:Qxz(z,x)→ 0, Qzz(z,x)→ p1(z), Qtt(z,x)→ p2(z), (D.3.3)Qyy(z,x)→ Qxx(z,x)→ p3(z) A0(z,x)→ A0(z), Φ(z,x)→Φ(z)Once the ODEs have been derived the horizon conditions can again be obtained inthe same manner described in Section 6.2 for the PDEs. In this case these boundaryconditions relate the value of the fields and their first radial derivative at the hori-zon. The conformal boundary conditions can also be written as a simple mixtureof Dirichlet and Neumann conditions as in the PDE case.p1(0) = 1, p2(0) = 1, p3(0) = 1, Φ′(0) = A, A0(0) = µ (D.3.4)The ODEs were solved via a simple adaption of the Chebychev grid spectraltechnique described in Section 6.4. The results also provided an additional sanitycheck on the solutions of PDE background equations Equation 6.2.2. When theseequations were solved with the sourced inhomogeneity turned off,Φ1(x) =A, goodagreement was obtained with the ODE solutions.229Considering the linearized perturbation equations around these ODE backgroundswe find that it is sufficient to restrict ourselves to perturbations of the followingform:hxt = h˜xt(z)e−iω t , bx = b˜x e−iω t (D.3.5)Utilizing the background equations of motion it can be checked that there are twoindependent equations of motion which can conveniently be decoupled from eachother and written as: (i) a first order equation for h˜xt in terms of the backgroundfields, and (ii) a second order equation for b˜x. We are only concerned with theequation for b˜x. Boundary conditions at the conformal boundary consist of b˜x(0) =1 corresponding to a conveniently normalized external electric field perturbation inthe dual QFT. At the horizon regularity in ingoing coordinates necessitates thefollowing leading scaling of the field b˜x =P(z) b˜regx . A suitable horizon boundarycondition may then be obtained via expanding b˜regx to leading order in (1− z).Once the solutions of the ODE equations were obtained we extracted the coef-ficient of the pole and calculated a goodness of fit measure given by(1− KhomKlat).The relevant results are displayed in percentage form in Table Table D.1. Klat is theresult obtained for the inhomogeneous sources.υ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.09 0.1Tµ =0.052109 0.57 0.26 1.08 1.89 2.69 3.48 4.26 5.78 6.53Tµ =0.027214 2.67 3.53 4.36 5.16 5.93 6.67 7.38 8.7 9.32Table D.1: Testing the fit to the Drude form of the conductivity and checkingagreement between the homogeneous and inhomogeneous solutions.230

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