1. AdS/CFT and the Fermions of condensed matter Jan Zaanen 1

2. The Gross list: the 14 Big Questions 2004 1. The origin of the universe? 2. What is dark matter? 11. What is space-time? 14. New states of matter: are there generic non-Fermi liquid states of interacting condensed matter? Solution of the Fermion sign problem??

3. ? Photoemission spectrum

4. 4 ‘Avatars of metallic phases’ Kachru et al., arXiv:0909.2639

5. 5 The ‘fermionic’ excitement (b) The emergent (heavy) Fermi liquid (Schalm, Leiden) (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (Liu, MIT) (d) Holographic superconductivity (Hartnoll, Harvard) (a) Hydrodynamics: ‘Planckian dissipation’ (Hartnoll, Sachdev, Harvard)

6. 6 Plan 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids. 1. Quantum critical electron systems and the great questions of condensed matter physics

7. 7 Quantum critical matter Quantum critical Heavy fermions High Tc superconductors Metamagnetic ruthenate Quark gluon plasma

8. 8 The quantum in the kitchen: Landau’s miracle Kinetic energy k=1/wavelength Electrons are waves Pauli exclusion principle: every state occupied by one electron Fermi momenta Fermi energy Fermi surface of copper Unreasonable: electrons strongly interact !! Landau’s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact.

9. 9 BCS theory: fermions turning into bosons Fermi-liquid + attractive interaction BardeenCooperSchrieffer Quasiparticles pair and Bose condense: D-wave SC: Dirac spectrumGround state

10. 10 ‘Shankar/Polchinski’ functional renormalization group interaction Fermi sphere UV: weakly interacting Fermi gas Integrate momentum shells: functions of running coupling constants All interactions (except marginal Hartree) irrelevant => Scaling limit might be perfectly ideal Fermi-gas

11. 11 The end of weak coupling interaction Fermi sphere Strong interactings: Fermi gas as UV starting point does not make sense! => ‘emergent’ Fermi liquid fixed point remarkably resilient (e.g. 3He) => Non Fermi-liquid/non ‘Hartree- Fock’ (BCS etc) states of fermion matter?

12. Fermion sign problem Imaginary time path-integral formulation Boltzmannons or Bosons:  integrand non-negative  probability of equivalent classical system: (crosslinked) ringpolymers Fermions:  negative Boltzmann weights  non probablistic: NP-hard problem (Troyer, Wiese)!!!

13. Fractal Cauliflower (romanesco)

14. Quantum critical cauliflower

18. 18 Quantum criticality or ‘conformal fields’

19. 19 Quantum Phase transitions Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations: JZ, Science 319, 1205 (2008)

20. 20 Fermionic renormalization group Wilson-Fisher RG: based on Boltzmannian statistical physics boundary: d-dim space-time Hawking radiation gluons Black holes strings quarks The Magic of AdS/CFT!

21. 21 The ‘fermionic’ excitement (b) The emergent (heavy) Fermi liquid (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (d) Holographic superconductivity (a) Hydrodynamics: ‘Planckian dissipation’

22. 22 Quantum critical matter Quantum critical Heavy fermions High Tc superconductors Metamagnetic ruthenate Quark gluon plasma

23. 23 Quantum criticality and Planckian dissipation Quark-gluon plasma: ‘minimal viscosity’ => absorption cross section of gravitons by the black hole (AdS/CFT) Quantum criticality => ‘Quantum limit of dissipation’: relaxation (= entropy production) time Linear resistivity in cuprates governed by ‘Planckian’ relaxation Van der Marel et al., Nature 425, 271 (2003); JZ, Nature 430, 512 (2004), Cooper et al., Science 323, 603 (2009)

24. 24 Dissipation = absorption of classical waves by Black hole! Viscosity: absorption cross section of gravitons by black hole Entropy density s: Bekenstein-Hawking BH entropy = area of horizon = area of horizon (GR theorems) Universal viscosity-entropy ratio for CFT’s with gravitational dual limited in large N by: Hartnoll-Son-Starinets (2002):

25. 25 The quark-gluon plasma Relativistic Heavy Ion ColliderQuark-gluon ‘fireball’

26. The tiny viscosity of the Quark- Gluon plasma QG plasma: within 20% of the AdS/CFT viscosity!

27. 27 Quantum critical hydrodynamics: Planckian relaxation time Planckian relaxation time = the shortest possible relaxation time under equilibrium conditions that can only be reached when the quantum dynamics is scale invariant !! Viscosity: “Planckian viscosity” Entropy density: Relaxation time : time it takes to convert work in entropy.

28. 28 Twenty three years ago … MuellerBednorz Ceramic CuO’s, likeYBa2Cu3O7 Superconductivity jumps to ‘high’ temperatures

29. 29 Graveyard of Theories Schrieffer Anderson Mueller Bednorz Laughlin Abrikosov Leggett Wilczek Mott Ginzburg De Gennes YangLee

30. 30 Phase diagram high Tc superconductors JZ, Science 315, 1372 (2007) Mystery quantum critical metal ‘Stripy stuff’, spontaneous currents, phase fluctuations.. The return of normalcy

31. 31 Divine resistivity

32. 32 Critical Cuprates are Planckian Dissipators A= 0.7: the normal state of optimallly doped cuprates is a Planckian dissipator ! van der Marel, JZ, … Nature 2003: Optical conductivity QC cuprates Frequency less than temperature:

33. 33 Divine resistivity ?!

34. 34 Planckian dissipation Quantum critical High Tc superconductors Quark gluon plasma Viscosity bound Planckian relaxation time: Linear resistivity normal state (JZ, Nature 430,512) : Tc is set by competition between superconductor and critical normal state (e.g. arXiv:0905.1225)

35. 35 Why the Wilsonian renormalization group fails … (a) Charge conservation (‘hydrodynamics’) imposes engineering scaling dimensions on current (b) Scale invariance: assume one diverging length scale. Phillips Chamon Superconductor- insulator QCP PRL 95, 107002 (2005) d =2 or 3 implies that z < 0 !?

36. 36 The ‘fermionic’ excitement (b) The emergent (heavy) Fermi liquid (c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (d) Holographic superconductivity (a) Hydrodynamics: ‘Planckian dissipation’

37. 37 Watching electrons: photoemission Kinetic energy k=1/wavelength Fermi momenta Fermi energy Fermi surface of copper Electron spectral function: probability to create or annihilate an electron at a given momentum and energy. k=1/wavelength Fermi energy energy

38. 38 Fermi-liquid phenomenology Bare single fermion propagator ‘enumerates the fixed point’: Spectral function: The Fermi liquid ‘lawyer list’: - At T= 0 the spectral weight is zero at the Fermi-energy except for the quasiparticle peak at the Fermi surface: - Analytical structure of the self-energy: - Temperature dependence:

39. 39 ARPES: Observing Fermi liquids ‘MDC’ at E F in conventional 2D metal (NbSe 2 ) Fermi-liquids: sharp Quasiparticle ‘poles’

40. 40 Cuprates: “Marginal” or “Critical” Fermi liquids Fermi ‘arcs’ (underdoped) closing to Fermi-surfaces (optimally-, overdoped). EDC lineshape: ‘branch cut’ (conformal), width propotional to energy

41. 41 Varma’s Marginal Fermi liquid phenomenology. Fermi- gas interacting by second order perturbation theory with ‘singular heat bath’: Directly observed in e.g. Raman ?? Single electron response (photoemission): Single particle life time is coincident (?!) with the transport life time => linear resistivity.

42. 42 Senthil’s critical Fermi-surface arXiv:0803.4092 Time like: truly conformal (branch cuts) Single particle spectral function: Space like: still a ‘critical’ Fermi surface (non conformal) Fermi-surface: higher order singularity in n(k)

43. 43 Critical fermions at zero density: branchcut propagators Two point Euclidean correlators: Analytically continue to Minkowski time => susceptibilities At criticality, conformal invariance: Lorentz invariance:Scaling dimension set by mass in AdS Dirac equation.

44. 44 Quantum critical matter Quantum critical Heavy fermions High Tc superconductors Metamagnetic ruthenate Quark gluon plasma

45. Quantum critical transport in heavy fermion systems Custers et al., Nature (2003) Blue = Fermi liquid: Yellow= quantum critical regime: Antiferromagnetic order

46. 46 Fermions and Hertz-Millis- Moriya-Lonzarich Fermi gas Bosonic (magnetic, etc.) order parameter drives the phase transition Electrons: fermion gas = heat bath damping bosonic critical fluctuations Bosonic critical fluctuations ‘back react’ as pairing glue on the electrons Supercon ductivity Ideologically like marginal FL:

47. Fermionic quantum phase transitions in the heavy fermion metals Paschen et al., Nature (2004) JZ, Science 319, 1205 (2008) QP effective mass ‘bad actors’ Coleman Rutgers

48. Critical Fermi surfaces in heavy fermion systems Blue = Fermi liquid Yellow= quantum critical regime Antiferromagnetic order FL Fermi surface Coexisting critical Fermi surfaces ?

49. 49 Plan 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids. 1. Quantum critical electron systems and the great questions of condensed matter physics

50. 50 SpielbergThorne HartnollHerzogHorowitz Fisk ThomsonRonning MacKenzie Grigeria Los AlamosSt Andrews Nature Nov 5 2009

51. 51 The zero temperature extensive entropy ‘disaster’ AdS-CFT The ‘extremal’ charged black hole in AdS has zero Hawking temperature but a finite horizon area. The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!)

52. 52 The holographic superconductor Hartnoll, Herzog, Horowitz, arXiv:0803.3295 (Scalar) matter ‘atmosphere’ AdS-CFT Condensate (superconductor, … ) on the boundary! ‘Super radiance’: in the presence of matter the extremal BH is unstable => zero T entropy always avoided by low T order!!!

53. 53 Hair as vacuum polarization

54. Fermionic quantum phase transitions in the heavy fermion metals Paschen et al., Nature (2004) JZ, Science 319, 1205 (2008) QP effective mass ‘bad actors’ Coleman Rutgers

55. 55 The metamagnetic quantum critical end point Metamagnetism: first order jump from low to high magnetization in field = transition between water and steam Ends at critical end point ‘St. Andrews’ ruthenate (Sr3Ru2O7): critical end point can be tuned to zero temperature GrigeraMackenzie

56. 56 The ‘entropic singularity’ (Rost et al., Science 325, 1360, 2009) QPT Zero T entropy ??

57. 57 The naked singularity always gets covered up … ‘Naked’ quantum critical point Exotic quantum nematic ‘hiding’ the critical point. ‘The unusual new phase can be thought of as the material’s solution to the problem of lowering its entropy in accord with the third law of thermodynamics’ Perspective Z. Fisk Science 325, 1348 (2009)

58. 58 Experimentalists: back to the entropic drawing board.. Grigeria MacKenzie ThomsonRonning Nailing down T=0 entropy hidden by last minute order: high precision entropy balance needed. Lanthanides, actinides: Los Alamos Ruthenates: St. Andrews

59. 59 Plan 2. Hairy black holes versus zero temperature entropy in quantum critical heavy fermion systems. 3. Fermion signs versus the AdS/CFT emergent (non) Fermi- liquids. 1. Quantum critical electron systems and the great questions of condensed matter physics

60. 60 “AdS-to-ARPES”: Fermi-liquid emerging from a quantum critical state. SchalmCubrovic

61. Fermionic quantum phase transitions in the heavy fermion metals Paschen et al., Nature (2004) JZ, Science 319, 1205 (2008) QP effective mass ‘bad actors’ Coleman Rutgers

62. 62 The fermionic criticality conundrum Kinetic energy k=1/wavelength Pauli exclusion principle generates the Fermi-energy, Fermi surface. Fermi momenta Fermi energy Fermi surface of copper How to reconcile the quantum statistical scales with scale invariance? How can a (heavy) Fermi-liquid emerge from a ‘microscopic’ quantum critical state? AdS/CFT gives an answer!

63. 63 Breaking fermionic criticality with a chemical potential Classical ‘Dirac waves’ Electrical monopole k E Fermi-surface??

64. 64 Fermion spectral function from RN Black hole (e.g. McGreevy, arXiv:0909.0518) Solve Dirac equations of motion, infalling boundary condition at Reissner- Nordstrom black hole horizon (horizon at z=1, boundary at z=0) Bulk Theory: Einstein-Maxwell + charged fermions Reissner-Nordstrom BH: 1. Universal: only depends on gq and m for any theory 2. Caveat: spectral function is a probe, possible dynamical instability ignored.

65. 65 “AdS-to-ARPES”: emergent Fermi-liquid. SchalmCubrovic QP peak! Emergent heavy Fermi liquid Scale invariance (branchcut) Lorentz + Scale invariance (branchcut):

66. 66 Fermi-liquid phenomenology Bare single fermion propagator ‘enumerates the fixed point’: Spectral function: The Fermi liquid ‘lawyer list’: - At T= 0 the spectral weight is zero at the Fermi-energy except for the quasiparticle peak at the Fermi surface: - Analytical structure of the self-energy: - Temperature dependence:

67. 67 “AdS-to-ARPES”: the quasiparticles. SchalmCubrovic Dip in the spectrum at

68. 68 “AdS-to-ARPES”: quasiparticle life times. SchalmCubrovic Temperature dependence: Energy dependence self-energy: No constant term in

69. 69 The AdS/CFT heavy Fermi liquid Landau Fermi-liquid Quasiparticle mass grows indefinitely when allowed by broken Lorentz invariance

70. 70 “AdS-to-ARPES”: summary. SchalmCubrovic AdS density Scaling dimension Fermion fields QP mass increases indefinitely in the emergent Fermi-liquid Fermionic quantum phase transition at finite density where the Fermi surface disappears

71. 71 Spectral function: overview Non-Fermi-liquid Landau Fermi-liquid 0

72. 72 The emergent horizon AdS 2 Horizon geometry of the extremal black hole: ‘emergent’ AdS 2 => IR of boundary theory controlled by emergent CFT 1 (Hong Liu) Gravitational ‘mechanism’ for marginal (critical) Fermi-liquids: Spatial: Fermi-surface implied by matching UV-IR Temporal: damping controlled by AdS 2, relevancy implies non Fermi-liquid Hong Liu (MIT)

73. 73 Gravitationally coding the fermion propagators (Faulkner et al. arXiv:0907.2694) T=0 extremal black hole, near horizon geometry ‘emergent scale invariant’: Matching with the UV infalling Dirac waves: Special momentum shell: Miracle, this is like critical/marginal Fermi-liquids!! Space-like: IR-UV matching ‘organizes’ Fermi-surface. Time-like: IR scale invariance picked up via AdS2 self energy

74. 74 The emergent horizon AdS 2 arXiv:0907.2694 CFT Fermion propagator: Fermi-liquids: ‘Critical Fermi liquids with Fermi surface: AdS 2

75. 75 Two families of emergent fermion infrareds Fermi-liquidEmergent “CFT 1 ” critical non-Fermi-liquids (reproduced in Leiden) “Marginal” Fermi-liquid “Pseudogap” phase see Liu et al., arXiv:0907.2694

76. 76 Thermodynamics: where are the fermions? Hartnoll et al.: arXiv:0908.2657,0912.0008 Large N limit: thermodynamics entirely determined by AdS geometry. Fermi surface dependent thermodynamics, e.g. Haas van Alphen oscillations? Leading 1/N corrections: “Fermionic one-loop dark energy” Quantum corrections: one loop using Dirac quasinormal modes: ‘generalized Lifshitz-Kosevich formula’ for HvA oscillations.

77. 77 Collective transport: fermion currents Tedious one loop calculation, ‘accidental’ cancellations: Hong Liu (MIT) ‘Strange coincidence’ of one electron and transport lifetime of marginal fermi liquid finds gravitational explanation!

78. 78 Soaked in Entropy …. Entropic catastrophe!

79. 79 The unstable RN Black hole: Fermion VEV’s! Witten: multi trace operators in AdS/CFT (hep-th/0112258) Fermion density J 0 could be finite !? Backreaction of matter: need VEV’s/classical finite amplitude fields in AdS. Back reaction on scalar potential A 0, keep RN-AdS geometry fixed: Cannot be formed from single fermion fields, but bilinears:

80. 80 The dictionary and Fermion VEV’s Following Witten’s proposal for multi trace operators in AdS/CFT (hep- th/0112258) How do bilinears like fit in the dictionary? The fields near the boundary (z -> 0): Proposal: The bilinears should go simply like the fields squared at the tree level

81. 81 Fermionic density back hole hair! Density switches on! First order gas-liquid transition at ‘high’ temperature: strongly interacting matter!

82. 82 Fermionic hair and the Fermi liquid stability. Strongly renormalized E F Single Fermion spectral function: pathological temperature dependent spectral weight transfer removed.

83. 83 The fermion probe limit as a lucky accident The change in BH electrical field is numerically very small But the field is diverging near the horizon The near-horizon geometry will backreact as well!

84. 84 Luttinger volume versus bulk density Luttinger theorem: the volume in k space enclosed by the Fermi surface of a Fermi liquid does not renormalize. Given k F (from the spectral functions) the density is: This is within numerical accuracy equal to the CFT density following from the bulk fermion VEV’s!

85. 85 Serious fermion hair: the Lifshitz horizon Divergence of E field at horizon: geometry has to backreact ! Solve equations of motion for Maxwell, Dirac and Einstein Metric: As for holographic superconductor this turns into a Lifshitz geometry (-> Horova): Lifshitz exponent

86. 86 Full backreaction: the stability of the Fermi-liquid! Non Fermi-liquid Fermi surfaces disappear from the spectral functions! Perfect Fermi-liquid

87. 87 Holographic superconductivity: stabilizing the fermions. Fermion spectrum for scalar-hair back hole (Faulkner et al., 911.340; Chen et al., 0911.282): ‘BCS’ Gap in fermion spectrum !! Temperature dependence as expected for ‘quantum-critical’ superconductivity (She, JZ, 0905.1225) Excessive temperature dependence ‘pacified’ !

88. 88 ‘Pseudogap’ fermions in high Tc superconductors 10 K Tc = 82 K 102 K Gap stays open above Tc But sharp quasiparticles disappear in incoherent ‘spectral smears’ in the metal Shen group, Nature 450, 81 (2007)

89. 89 The moral of this story… The AdS/CFT correspondence: processing the fermion signs in fermionic quantum critical states The mystery of Fermi-liquid stability: in the ‘hairy’ gravitational dual it is just like a bosonic condensate!! Critical Fermi-liquid phenomenology: AdS 2 horizons needed, do they occur at ‘isolated’ quantum critical points associated with bosonic order? Holographic superconductors feeds on the AdS 2 T=0 entropy: employment program for condensed matter experimentalists!

90. 90 Further reading AdS/CMT tutorials: J. Mc Greevy, arXiv:0909.0518; S. Hartnoll, arXiv:0909.3553 AdS/CMT fermions: Hong Liu et al., arXiv:0903.2477,0907.2694; K. Schalm et al. Science, and to appear any moment; T. Faulkner et al., arXiv: 0911.3402 Condensed matter: High Tc: J. Zaanen et al., Nature 430, 512, Nature Physics 2, 138; C.M. Varma et al., Phys. Rep. 361, 267417 Heavy Fermions: J. Zaanen, Science 319, 1205; von Loehneisen et al, Rev. Mod. Phys. 79, 1015

91. 91 Thermodynamics from classical action Total action: Bulk action: equation of motions Boundary action For fermions bulk action vanishes in the classical limit But boundary action is finite => Free energy is determined by I 0 and J 0

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