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# Index

1*/N* corrections, 28, 53, 108–110, 161–166, 282–284, 311

AdS radius, 11, 17, 29, 31, 34–35

AdS soliton, 69

AdS-Reissner-Nordström black hole, 127–128

spatial modulation, 291–292

AdS-Schwarzchild black hole, 61

AdS_{2} × ℝ^{d}, 129–131

comparison to SYK, 253

fermionic instability, 167–170

generalized umklapp, 222

inhomogeneous fixed points, 238

operator spectral weight, 140–143

translation symmetry breaking, 290

AdS_{5} × S^{5}, 11, 12, 34, 36–38, 304–305

algebraic long range order, 282–284

alternate quantization, 27–28, 267

anomalous dimension, 44

energy density, 56

anomaly, 176

magnetotransport, 230–232

antiferromagnetism, 46–49

BCS instability, 117, 164, 266

Bekenstein-Hawking formula, 5

at zero temperature, 128

disconnected horizons, 250–251

entropy, 61

formation, 319–321

inhomogeneous, 248–251, 292–295

scalar hair, 267–268

superfluid flow, 280–281

Bloch’s law, 185

Boltzmann equation, 77

bottom-up holography, 35

boundary conditions at horizon

at zero temperature, 86–87, 136

infalling, 6, 19, 82–83, 90–92, 94, 99, 103–104

bounds, 334–335

diffusion, 234–235

viscosity, 234

Breitenlohner-Freedman bound, 267, 271

bulk fermion, 151–153

backreaction on spacetime, 166–171

consistent truncations, 38, 155

contribution to charge, 158–160

decay rate, 154–155

gap in superconductor, 284–286

mass and operator dimension, 153

pair production, 167

relation to conductivity, 157

semi-holographic, 155–158

with dipole coupling, 155

bulk fields

boundary conditions, 19

multi trace operators, 27

equations of motion, 133

number of, 17–18

operator dual, 20–21

bulk gauge field, 20

constant electric field, 89

coupling to Weyl curvature, 95

DBI action, 305

electromagnetic duality, 100, 105

gauge transformation, 20

non-Abelian, 286–287

bulk partition function, 13

divergences, 22–23

bulk scalar, 15–17

AdS solution, 19

AdS_{2} × ℝ^{d} solution, 140–141

and hyperscaling violation, 137

boundary conditions, 18

charged, 268

mass and operator dimension, 16, 55

nonzero temperature, 145–149

bulk spin two field

charged, 288–289

butterfly velocity, 234–235

and non-dissipative flow, 201–202

bulk electric flux, 125

thermodynamic relations, 115

for bulk fermions, 167–169

cohesive charge, 123–126, 158, 169–170

Fermi surfaces, 170–171

from Chern-Simons term, 176–177

versus confinement, 275

competing orders, 286

compressibility, *see* charge susceptibility

compressible matter

absence of gap, 114

field theories, 115–124

conductivity, 74, 89, 93, 189, 209

alternating current, 241–242, 249–250

and quasinormal modes, 105, 106

anomalous scaling, 138

at weak disorder, 203, 214, 216

at zero density, 72–81, 87–102

bulk fermions, 165–166

density waves, 295–296

direct current, 89–91, 238, 244–245

from horizon fluid, 244–245

from memory matrix, 211

graphene, 71

holographic formula, 94

in Einstein-Maxwell theory, 94

incoherent, 138, 198, 204, 215

nonlinear, 312–313

numerical computation, 97–98, 250

of free Dirac fermions, 76–77

particle-vortex duality, 75, 100–102, 174

probe brane models, 311–314

residual, 184

short time asymptotics, 78–79, 96, 110

striped inhomogeneity, 246

superconductivity, 276–280

with Weyl curvature coupling, 97

and superconductivity, 275

conformal field theory, 15

integrability of CFT2, 73

scalar three-point function, 39

scalar two-point function, 20

conformal group, 15

consistent truncation, 35–38

constitutive relations, 193, 201, 232

Cooper pairing, 163–165, 266, 273, 279

correlation length, 45

finite temperature, 62–64

counterterm, 23

critical temperature, 269, 272, 275, 298

normalizable mode, 271, 273, 300

crystal, 294–295

Hall angle, 230

correlators, 73–74

in CFT2, 72

cyclotron resonance, 175, 228–230

intersections, 304

tension, 305

*d*-wave superconductivity, 288–290

spin density wave point, 122, 266

deconfinement, 158

density wave, 292–294

hydrodynamic Green’s function, 73

diffusion constant, 74

experimental measurement, 337

UV sensitivity, 93

dilaton, 131

dimerization, 315

Dirac-Born-Infeld action, 305

disorder, 186, 204, 222, 315–318

disordered fixed point, 315–318, 328

domain wall spacetime, 65

double trace operator, 27, 297

Drude peak, 77, 195–197, 199–200, 207, 211–214

holographic derivation, 223–227

in magnetic field, 228

loss at strong disorder, 242

mean-field disorder, 238

probe brane models, 313–314

superconductor, 277–278

weight, 199–200

dynamic critical exponent, 44, 53–56, 58, 120, 220, 275, 323, 332

holographic superconductor, 275

dynamical gauge field, 277

Eddington-Finkelstein coordinates, 90

Einstein-Hilbert action, 14

Einstein-Maxwell theory, 36–38, 87, 126

and charged scalars, 36–37

and neutral scalars, 37–38

charged black holes, 126–131

conductivity in *d* = 2, 94, 101–102

topological term, 174–175

with Chern-Simons term, 36, 176–178, 290–292

Einstein-Maxwell-dilaton theory, 37, 54, 128, 131–137, 149

anomalous charge density, 138

Einstein-Proca theory, 54

emergent gauge field, 49–52, 122–124

energy density, 191

relation to pressure, 115

entanglement entropy, 29–31

area law, 30

logarithmic violation, 31

Fermi surface, 150–151

entropy balance, 275

entropy density, 61, 67, 114, 136, 137, 151, 178, 192, 216

finite at *T* = 0, 128, 253, 308

thermodynamic relations, 115

entropy production, 84

hydrodynamic constraints, 201–202

dissipation, 2, 5–7, 83–85, 166, 318, 325

entropy production, 83

extremal black hole, 128

ground state degeneracy, 131, 253

Fermi arc, 286

Fermi liquid, 115–118

effective action, 117

holographic, 160

scaling geometries, 144–145

Fermi surface, 116

*α*^{′} corrections, 145

bulk monopoles, 145

entanglement entropy, 150–151

holographic, 157–158

hyperscaling violation, 115, 150–151

pole in Green’s function, 154–155

first order transition, 248, 270, 281, 291, 293

fluid-gravity correspondence, 193, 258

fractionalization, 122–124, 138, 158

bosons, 274

finite temperature, 170

gapped theory, 64–67

*p*-wave superconductivity, 287–288

gauge invariant variables, 88, 143

generating functional, 13

geodesic, 33, 40, 64, 66, 68, 250

Gibbs-Duhem identity, 115

GKPW formula, 12–14

Goldstone mode, 28, 44, 103, 108, 277, 280, 282, 295, 298, 321

Gross-Neveu model, 47

Gubser-Rocha black hole, 299

Hall angle, 230

Hall conductivity, 172–173

*p*-wave superconductivity, 287

hard wall geometry, 160

Harris criterion, 316–317

Hawking-Page transition, 67, 70

heat current, 187

as entropy current, 193

heavy fermion metals, 338

instability towards, 290–292

high *T*_{c} superconductivity, 3, 266

higher derivative theory

bounds on coefficients, 96

bulk gradient expansion, 17, 95

conductivity, 95–98

holographic dictionary, 12–21

UV cutoff, 24–26

holographic duality, 9–12

extra compact dimensions, 34

non-geometrized degrees of freedom, 28

symmetry, 20

holographic lattice, 220–222, 248–249

holographic renormalization, 23

Hubbard model

on honeycomb lattice, 46–49

on square lattice, 43–46

hydrodynamics, 190–195

breakdown, *see* long-time tails

Galilean, 202–203

general linear response, 200–202

Green’s function, 197

quasinormal modes, 106–107, 194–195

relativistic, 191–195

superfluid, 281–282

transport, 186, 203–207, 330–331

hyperscaling violating geometry, 56–59, 134

regular, 136

singularities, 58

hyperscaling violation exponent, 57, 114, 138, 217, 332

incoherent diffusion, 194, 198–199

from memory matrix, 214–217

in condensed matter, 217

incoherent metal, 186, 232–233

instability, 154, 155, 167, 222, 267–270, 272, 281, 286, 290–296, 300, 333

Ising-nematic criticality, 118–120

conductivity, 218–219

incoherent conductivity, 217

superconducting instability, 266

viscosity, 219

Josephson junction, 276

Joule heating, 173

Kaluza-Klein mode, 17, 34–35, 39

and gapped theories, 66

spin two modes, 289–290

Kibble-Zurek mechanism, 323–324

Kohn’s theorem, 229–230

Kondo model, 338

holographic, 314–315

Landau frame, 192

Landau-Ginzburg theory, 275

large *N* limit, 7–9

breakdown, 28

dual semiclassical description, 18

matrix theory, 10

thermodynamics, 62

Lifshitz geometry, 53–56

double trace deformation, 88–89

electron star, 169–170

from string theory, 54

infalling boundary condition, 86

marginal disorder, 317–318

singularities, 53

spectral weight, 148–149

superconducting, 274

Lifshitz-Kosevich formula, 161, 332

linear axion model, 236–238

linear response, 81–83

local criticality, 128–129, 145, 331–332

conductivity, 222–223

continuum of operator dimensions, 141, 290

critical phase, 131

insulating instability, 239–240

phenomenology, 332

spatial modulation, 293

strong inhomogeneity, 249

logarithmic running, 86

Luttinger’s theorem, 116

holographic, 158–160

with topological order, 124

zeros vs. poles, 116

Lyapunov time, 4

magnetic field, 102, 161, 171–178, 227, 229, 265, 275, 276, 292–294, 298, 313

magnetotransport, 175–176, 227–230

many-body localization, 247, 251

marginal Fermi liquid, 155

massive gravity, 236

matching argument, 141–143, 327

matrix product state, 32

mean free path, 184

mean-field disorder, 203, 236–239

comparison to SYK, 258

transport bounds, 247

Meissner effect, 276

membrane paradigm, 6–7

conductivity, 89–91

memory matrix, 207–211

holographic conductivities from, 220–223

holographic derivation, 223–227

in condensed matter, 218–219

magnetic field, 228–229

MERA, 33

metal-insulator transition, 233, 239–241

in inhomogeneous system, 249

momentum relaxation time, 203–204

from hydrodynamics, 204–207

Mott insulator, 43

Mott-Ioffe-Regel bound, 184, 233–234

multi-trace operator, 21, 27–28

mutual information, 40

𝒩 = 4 super Yang-Mills theory, 9–12, 36, 37, 62, 127, 304–305

negative magnetoresistance, 231

Newton’s constant, 15

no hair theorem, 268

non-critical string theory, 34

non-equilibrium steady state, 321–322

null energy condition, 57, 136

operator

inner product, 200

scaling dimension, 16

source, 19

operator product expansion, 78

*p*-wave superconductivity, 286–288

pair density waves, 286

particle-vortex duality, 75, 100–102, 105

compact boson, 100

disorder, 248

higher derivative theory, 97

probe brane models, 314

self-dual theories, 100

phonon drag, 185

pnictides, 337–338

Poincaré horizon, 15

pressure, 114

bulk fermions, 168

probe brane, 304–307

conductivity, 311–314

embedding, 307–308

supersymmetric configuration, 305

thermodynamics, 308

Q-lattice, 236

quadratic band touching, 49

quantum critical conductivity, 193, 197–198

quantum critical phase, 95

quantum critical point

relevant operator, 45, 64–65, 98, 239

quantum criticality, 42–52, 332–333

quantum Monte Carlo, 80–81

quantum oscillations, 161–163, 171, 265

quark, 304

quasinormal mode, 2, 98, 99, 102–107, 111, 310

analytic computation, 111

diffusion, 91–93

in quench, 320–321

instability, 273

numerical computation, 103–104

versus branch cuts, 103

WKB computation, 104

quasiparticle, 3

and ballistic transport, 77

away from quantum criticality, 46

holographic, 154

in Fermi liquid, 116

lifetime, 4

quench, 318

radial gauge, 87

random-field disorder, 222–223

renormalization group

beta function, 26

bulk dimension, 21–26

resistivity saturation, 238

resistor network theory, 233

resonating valence bond model, 49–52

retarded Green’s function, 20, 81–83

causality, 103

diffusive pole, 91–93

from Euclidean Green’s function, 82

quasinormal poles, 103

thermoelectric conductivities, 189

Ryu-Takayanagi formula, 29

S-duality, *see* particle-vortex duality

Sachdev-Ye-Kitaev model, 4, 9, 251–253

effective action, 254–256

spatial dimensions, 256–258

scaling geometry, 52–58, 134–139

stringy/quantum corrections, 53, 135–136

UV cutoff, 135

Schrödinger geometry, 58–59, 68, 177

Schrödinger scattering problem, 83

Schutz action, 167

Schwarzian action, 255–256

second law of thermodynamics, 192

second sound, 282–283

Seebeck coefficient, 190

semi-holography, 155–158

BKT transition, 298

diffusion, 93

superconductivity, 284

semi-local criticality, *see* local criticality

single trace operator, 8, 17, 21

holographic Green’s function, 162

sound waves, 107, 193–195, 198, 259, 310, 330

gapped system, 67

superconductor, 274

and conductivity, 220

at nonzero momentum, 141–144, 309–310

Fermi surface non-analyticity, 140

fields on horizon, 145–147

large dimension operator, 83–86

Lifshitz geometry, 55–56

low energy, 143

nonzero temperature, 145–149

probe brane models, 309–310

zero temperature, 140–145

spectral weight transfer, 204

spin density wave criticality, 120–122

conductivity, 219

incoherent conductivity, 217

superconducting instability, 266

spin-orbit coupling, 155

static susceptibility, 197

relativistic hydrodynamics, 200

string coupling constant, 9

string length, 11

striped phase, *see* density wave

fermions, 155

superconductor, 279–280

superconductivity, 273–276

from holographic fermions, 165

holographic instability, 270–273

Ising-nematic instability, 120

non-Fermi liquids, 266–267

supercurrent, 200, 277, 280, 281

quench, 320

versus superconductor, 277

superfluid turbulence, 325–327

*T*-linear resistivity, 337, 338

and horizon radius, 61

and periodic Euclidean time, 59

screening, 62–64

temperature gradient, 188

tensor network, 31

thermal conductivity, 190, 198, 202

thermalization, 7

absence, 319

instantaneous, 320

thermodynamics, 114–115

low temperature scaling, 129, 137

superfluid, 280

thermoelectric conductivity, 187–190

mean-field disorder, 242–243

SYK model, 257–258

translation invariant, 202

thermofield double, 83

Thomas-Fermi approximation, 166–171

top-down holography, 35

topological order, 122

in cuprates, 337

in heavy fermion metals, 338

turbulence, 324–325

umklapp, 184, 185, 221, 222, 239

unitarity bound, 267

Villain model, 80–81

Fermi liquid, 336

in condensed matter, 219

vortex, 100

annihilation rate, 326–327

formation, 323–324

superconductivity, 276

vortex lattice, 294–295

Ward identity, 137, 197, 203, 205

with disorder, 205

Weyl semimetal, 230–231

in holography, 231–232

Wiedemann-Franz law, 183, 184, 198, 330–331, 335

Wilson-Fisher fixed point, 44

WKB approximation, 55, 63–64, 84–85, 96, 104, 149, 158, 159, 166, 309

Wronskian method, 145–147