Holographic Quantum Matter

by Hartnoll, Lucas, Sachdev

ISBN: 9780262348003 | Copyright 2018

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Contents (pg. vii)
Preface (pg. xiii)
1. The holographic correspondence (pg. 1)
1.1. Historical context I: Quantum matter without quasiparticles (pg. 2)
1.2. Historical context II: Horizons are dissipative (pg. 4)
1.3. Historical context III: The 't Hooft matrix large N limit (pg. 6)
1.4. Maldacena's argument and the canonical examples (pg. 8)
1.5. The essential dictionary (pg. 11)
1.5.1. The GKPW formula (pg. 11)
1.5.2. Fields in AdS spacetime (pg. 13)
1.5.3. Simplification in the limit of strong QFT coupling (pg. 16)
1.5.4. Expectation values and Green's functions (pg. 17)
1.5.5. Bulk gauge symmetries are global symmetries of the dual QFT (pg. 19)
1.6. The emergent dimension I: Wilsonian holographic renormalization (pg. 20)
1.6.1. Bulk volume divergences and boundary counterterms (pg. 21)
1.6.2. Wilsonian renormalization as the Hamilton-Jacobi equation (pg. 22)
1.6.3. Multi-trace operators (pg. 26)
1.6.4. Geometrized versus non-geometrized low energy degrees of freedom (pg. 27)
1.7. The emergent dimension II: Entanglement entropy (pg. 27)
1.7.1. Analogy with tensor networks (pg. 30)
1.8. Microscopics: Kaluza-Klein modes and consistent truncations (pg. 33)
Exercises (pg. 37)
2. Zero density matter (pg. 41)
2.1. Condensed matter systems (pg. 42)
2.1.1. Antiferromagnetism on the honeycomb lattice (pg. 46)
2.1.2. Quadratic band-touching and z ≠ 1 (pg. 49)
2.1.3. Emergent gauge fields (pg. 49)
2.2. Scale invariant geometries (pg. 52)
2.2.1. Dynamic critical exponent z > 1 (pg. 53)
2.2.2. Hyperscaling violation (pg. 56)
2.2.3. Galilean-invariant 'non-relativistic CFTs' (pg. 58)
2.3. Nonzero temperature (pg. 59)
2.3.1. Thermodynamics (pg. 59)
2.3.2. Thermal screening (pg. 62)
2.4. Theories with a mass gap (pg. 64)
Exercises (pg. 67)
3. Quantum critical transport (pg. 71)
3.1. Condensed matter systems and questions (pg. 72)
3.2. Standard approaches and their limitations (pg. 75)
3.2.1. Quasiparticle-based methods (pg. 75)
3.2.2. Short time expansion (pg. 78)
3.2.3. Quantum Monte Carlo (pg. 80)
3.3. Holographic spectral functions (pg. 81)
3.3.1. Infalling boundary conditions at the horizon (pg. 82)
3.3.2. Example: spectral weight Im GROO(w) of a large dimension operator (pg. 83)
3.3.3. Infalling boundary conditions at zero temperature (pg. 86)
3.4. Quantum critical charge dynamics (pg. 87)
3.4.1. Conductivity from the dynamics of a bulk Maxwell field (pg. 87)
3.4.2. The dc conductivity (pg. 89)
3.4.3. Diffusive limit (pg. 91)
3.4.4. (w) part I: Critical phases (pg. 93)
3.4.5. (w) part II: Critical points and holographic analytic continuation (pg. 98)
3.4.6. Particle-vortex duality and Maxwell duality (pg. 100)
3.5. Quasinormal modes replace quasiparticles (pg. 102)
3.5.1. Physics and computation of quasinormal modes (pg. 102)
3.5.2. 1/N corrections from quasinormal modes (pg. 108)
Exercises (pg. 110)
4. Compressible quantum matter (pg. 113)
4.1. Thermodynamics of compressible matter (pg. 114)
4.2. Condensed matter systems (pg. 115)
4.2.1. Ising-nematic transition (pg. 118)
4.2.2. Spin density wave transition (pg. 120)
4.2.3. Emergent gauge fields (pg. 122)
4.3. Charged horizons (pg. 125)
4.3.1. Einstein-Maxwell theory and AdS2 x Rd (or, z = ∞) (pg. 126)
4.3.2. Einstein-Maxwell-dilaton models (pg. 131)
4.3.3. Critical compressible phases with diverse z and θ (pg. 134)
4.3.4. Anomalous scaling of charge density (pg. 138)
4.4. Low energy spectrum of excitations (pg. 139)
4.4.1. Spectral weight: zero temperature (pg. 140)
4.4.2. Spectral weight: nonzero temperature (pg. 145)
4.4.3. Logarithmic violation of the area law of entanglement (pg. 150)
4.5. Fermions in the bulk I: 'Classical' physics (pg. 151)
4.5.1. The holographic dictionary (pg. 152)
4.5.2. Fermions in semi-locally critical (z = ∞) backgrounds (pg. 153)
4.5.3. Semi-holography: One fermion decaying into a large N bath (pg. 155)
4.6. Fermions in the bulk II: Quantum effects (pg. 158)
4.6.1. Luttinger's theorem in holography (pg. 158)
4.6.2. 1/N corrections (pg. 161)
4.6.3. Endpoint of the near-horizon instability in the fluid approximation (pg. 166)
4.7. Magnetic fields (pg. 171)
4.7.1. d = 2: Hall transport and duality (pg. 172)
4.7.2. d = 3: Chern-Simons term and quantum phase transition (pg. 176)
Exercises (pg. 178)
5. Metallic transport without quasiparticles (pg. 183)
5.1. Metallic transport with quasiparticles (pg. 184)
5.2. The momentum bottleneck (pg. 184)
5.3. Thermoelectric conductivity matrix (pg. 187)
5.4. Hydrodynamic transport (with momentum) (pg. 190)
5.4.1. Relativistic hydrodynamics near quantum criticality (pg. 191)
5.4.2. Sound waves (pg. 193)
5.4.3. Transport coefficients (pg. 195)
5.4.4. Drude weights and conserved quantities (pg. 199)
5.4.5. General linearized hydrodynamics (pg. 200)
5.5. Weak momentum relaxation I: Inhomogeneous hydrodynamics (pg. 203)
5.6. Weak momentum relaxation II: The memory matrix formalism (pg. 207)
5.6.1. The Drude conductivities (pg. 211)
5.6.2. The incoherent conductivities (pg. 214)
5.6.3. Transport in field-theoretic condensed matter models (pg. 217)
5.6.4. Transport in holographic compressible phases (pg. 220)
5.6.5. From holography to memory matrices (pg. 223)
5.7. Magnetotransport (pg. 227)
5.7.1. Weyl semimetals: Anomalies and magnetotransport (pg. 230)
5.8. Hydrodynamic transport (without momentum) (pg. 232)
5.9. Strong momentum relaxation I: 'Mean-field' methods (pg. 236)
5.9.1. Metal-insulator transitions (pg. 239)
5.9.2. AC transport (pg. 241)
5.9.3. Thermoelectric conductivities (pg. 242)
5.10. Strong momentum relaxation II: Exact methods (pg. 244)
5.10.1. Analytic methods (pg. 244)
5.10.2. Numerical methods (pg. 248)
5.11. SYK models (pg. 251)
5.11.1. Fluctuations (pg. 253)
5.11.2. Higher dimensional models (pg. 256)
Exercises (pg. 258)
6. Symmetry broken phases (pg. 265)
6.1. Condensed matter systems (pg. 266)
6.2. The Breitenlohner-Freedman bound and IR instabilities (pg. 267)
6.3. Holographic superconductivity (pg. 270)
6.3.1. The phase transition (pg. 270)
6.3.2. The condensed phase (pg. 273)
6.4. Response functions in the ordered phase (pg. 276)
6.4.1. Conductivity (pg. 276)
6.4.2. Superfluid hydrodynamics (pg. 280)
6.4.3. Destruction of long range order in low dimension (pg. 282)
6.4.4. Fermions (pg. 284)
6.5. Beyond charged scalars (pg. 286)
6.5.1. Homogeneous phases (pg. 286)
6.5.2. Spontaneous breaking of translation symmetry (pg. 290)
6.6. Zero temperature BKT transitions (pg. 296)
Exercises (pg. 299)
7. Further topics (pg. 303)
7.1. Probe branes (pg. 304)
7.1.1. Microscopics and effective bulk action (pg. 304)
7.1.2. Backgrounds (pg. 307)
7.1.3. Spectral weight at nonzero momentum and 'zero sound' (pg. 308)
7.1.4. Linear and nonlinear conductivity (pg. 311)
7.1.5. Defects and impurities (pg. 314)
7.2. Disordered fixed points (pg. 315)
7.3. Out of equilibrium I: Quenches (pg. 318)
7.3.1. Uniform quenches (pg. 319)
7.3.2. Spatial quenches (pg. 321)
7.3.3. Kibble-Zurek mechanism and beyond (pg. 323)
7.4. Out of equilibrium II: Turbulence (pg. 324)
Exercises (pg. 327)
8. Connections to experiments (pg. 329)
8.1. Probing non-quasiparticle physics (pg. 330)
8.1.1. Parametrizing hydrodynamics (pg. 330)
8.1.2. Parametrizing low energy spectral weight (pg. 331)
8.1.3. Parametrizing quantum criticality (pg. 332)
8.1.4. Ordered phases and insulators (pg. 333)
8.1.5. Fundamental bounds on transport (pg. 334)
8.2. Experimental realizations of strange metals (pg. 335)
8.2.1. Graphene (pg. 335)
8.2.2. Cuprates (pg. 336)
8.2.3. Pnictides (pg. 337)
8.2.4. Heavy fermions (pg. 338)
Bibliography (pg. 339)
Index (pg. 383)

Sean A. Hartnoll

Sean A. Hartnoll is Associate Professor of Physics at Stanford University and was the recipient of the New Horizons Prize in Physics in 2015.


Andrew Lucas

Andrew Lucas is a Gordon and Betty Moore Fellow in Theoretical Condensed Matter Physics at Stanford University.


Subir Sachdev

Subir Sachdev is Herchel Smith Professor of Physics at Harvard University and the author of Quantum Phase Transitions. He was awarded the Lars Onsager Prize from the American Physical Society in 2018.


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