StringString theorytheory andand thethe mysteriousmysterious quantumquantum mattermatter ofof condensedcondensed mattermatter physics.physics.

Jan Zaanen

1 String theory: what is it really good for?

- Hadron (nuclear) physics: quark-gluon plasma in RIHC. - Quantum matter: quantum criticality in heavy fermion systems, high Tc superconductors, …

Started in 2001, got on steam in 2007.

Son Hartnoll Herzog Kovtun McGreevy Liu Schalm

2 Quantum critical matter

Quark gluon plasma

Iron High Tc Heavy fermions superconductors superconductors (?)

Quantum critical Quantum critical

3 High-Tc Has Changed Landscape of Condensed Matter Physics High-resolution ARPES Magneto-optics Transport-Nernst effect

Spin-polarized Neutron STM High Tc Superconductivity

Inelastic X-Ray Scattering Angle-resolved MR/Heat Capacity ?

Photoemission spectrum Hairy Black holes …

6 Holography and quantum matter

But first: crash course in holography

“Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like overdoped high Tc (Schalm, Cubrovic, Hartnoll).

Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

7 General relativity “=“ quantum field theory

Gravity Quantum fields

Maldacena 1997 =

8 Anti de Sitter-conformal quantum field theory correspondence

AdS geometry Conformal quantum field (“near” the boundary) theory (at ‘high’ energies)

Not like our universe … Another word for: Quantum criticality! 9 Holography with lasers

Encoded on a two Three dimensional image dimensional photographic plate Gravity - quantum field holography

Einstein world “AdS” = Quantum fields in flat space Anti de Sitter universe “CFT”= quantum critical Hawking radiation ‘t Hooft’s holographic principle

Hawking Temperature:

g = acceleration at horizon

BH entropy: 0 1 0 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 1 A = area of horizon 0 0 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 Number of degrees of freedom (field 0 1 1 0 1 1 0 0 theory) scales with the area and not 0 1 0 0 1 1 1 1 1 0 0 1 with the volume (gravity) 0 1 The bulk: Anti-de Sitter space

Extra radial dimension of the bulk <=> scaling “dimension” in the field IR UV theory

Bulk AdS geometry = scale invariance of the field theory Fractal Cauliflower (romanesco) Quantum critical cauliflower Quantum critical cauliflower Quantum critical cauliflower Quantum critical cauliflower Fermion sign problem

Imaginary time path-integral formulation

Boltzmannons or Bosons: Fermions: . integrand non-negative . negative Boltzmann weights . probability of equivalent classical . non probablistic: NP-hard system: (crosslinked) ringpolymers problem (Troyer, Wiese)!!! Renormalization group for quantum critical matter

The Magic of AdS/CFT! boundary: strings d-dim space-time Black holes

Wilson-Fisher RG: based on Boltzmannian statistical physics

gluons quarks Hawking radiation 21 Black hole hair codes the quantum matter

“Hairy black holes” code for (un)stable states of quantum matter emerging from the quantum critical stuff.

22 Quantum critical dynamics: classical waves in AdS

WCFT (J) = SAdS (φ) 0 J φx0 → = R4 g2 N = YM α 2 gYM = gs

€ 23 The AdS/CFT dictionary

SUSY Einstein-Maxwell in AdS <==> SUSY Yang-Mills CFT

E-field transverse E-field <=> 3d electric field radial E-field <=> 3d charge density

B-field radial B-field <=> 3d magnetic field transverse B-field <=> 3d current density

spatial metric perturb. transverse gradient <=> 3d distortion radial gradient <=> 3d stress tensor

temporal metric perturb. transverse gradient <=> temperature gradient radial gradient <=> heat flow Holography and quantum matter

But first: crash course in holography

“Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like overdoped high Tc (Schalm, Cubrovic, Hartnoll).

Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

25 26 The Schwarzschild Black Hole is the heater

Black hole temperature entropy

GR in Anti de Sitter space Quantum-critical fields on the boundary: - at the Hawking temperature - entropy = black hole entropy 27 Dissipation = absorption of classical waves by Black hole!

Policastro-Son-Starinets (2002): Viscosity: absorption cross section of

gravitons by black hole σ abs(0) η = 16πG = area of horizon (GR theorems)

Entropy density s: Bekenstein-Hawking € BH entropy = area of horizon

Universal viscosity-entropy ratio for CFT’s η 1 h with gravitational dual limited in large N by: = s 4π kB 28

€ AdS/CFT viscosity Kovtun-Son-Starinets (2005)

4πkBη hs

€ The quark-gluon plasma

Relativistic Heavy Ion Collider Quark-gluon ‘fireball’

30 The tiny viscosity of the Quark- Gluon plasma

4πkBη hs QG plasma: within€ 20% of the AdS/CFT viscosity! Quantum critical hydrodynamics: Planckian dissipation & viscosity 1 Planckian dissipation: ω In a finite temperature quantum critical state the time it takes to convert work in h heat (relaxation time) has to be k T B €

Sachdev, h 1992 τ = τ ≈ h € kBT ε + p η Viscosity, entropy density: η = (ε + p)τ, s = ⇒ = Tτ T s η Planckian viscosity: ≈ h s kB € € 32

€ Twenty five years ago …

Mueller Bednorz Ceramic CuO’s, likeYBa2Cu3O7

Superconductivity jumps to ‘high’ temperatures

33 Graveyard of Theories

Mueller Schrieffer Mott Laughlin

Abrikosov De Gennes Bednorz Anderson Leggett

Lee Wilczek Ginzburg Yang 34 Phase diagram high Tc superconductors Mystery ‘Stripy stuff’, spontaneous quantum critical currents, phase fluctuations .. metal The return of normalcy

+ + ΨBCS = Πk (uk + vkck↑c−k↓) vac. JZ, Science 315, 1372 (2007) € 35 Quantum Phase transitions

Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations:

JZ, Science 319, 1205 (2008) 36 A universal phase diagram

High Tc Heavy fermions Iron superconductors superconductors (?)

Quantum critical Quantum critical

37 Divine resistivity

38 Critical Cuprates are Planckian Dissipators van der Marel, JZ, … Nature 2003:

Optical conductivity QC cuprates

Frequency less than temperature: 2 1 ω prτ r h σ1(ω,T) = 2 2 , τ r = A 4π 1+ ω τ r kBT ω ⇒ [ h ] = const.(1+ A2[ h ]2) k Tσ k T B 1 B € A= 0.7: the normal state of optimallly doped cuprates is a Planckian dissipator! € 39 Divine resistivity = Planckian Dissipation!

1 ρ ∝ ∝ k T τ B h

€

40 Holography and quantum matter

But first: crash course in holography

“Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll).

Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

41 Holographic quantum critical fermion state Liu McGreevy

42 The quantum in the kitchen: Landau’s miracle Kinetic energy Electrons are waves

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

43 Watching electrons: photoemission Kinetic energy Electron spectral function: probability to create or annihilate an electron at a given momentum and energy. Fermi energy

Fermi momenta Fermi k=1/wavelength energy

energy

Fermi surface of copper

k=1/wavelength 44 ARPES: Observing Fermi liquids

‘MDC’ at EF in conventional Fermi-liquids: sharp Quasiparticle ‘poles’ 2D metal (NbSe2)

45 Cuprates: “Marginal” or “Critical” Fermi liquids

Fermi ‘arcs’ (underdoped) closing to Fermi-surfaces EDC lineshape: ‘branch cut’ (conformal), (optimally-, overdoped). width propotional to energy 46 Breaking fermionic criticality with a chemical potential

‘Dirac waves’ E

Fermi-surface?? µ

µ k

Electrical monopole€ € 47 AdS/ARPES for the Reissner- Nordstrom non-Fermi liquids

Fermi surfaces but no quasiparticles!

Critical FL Marginal FL Non Landau FL

48 Gravitationally coding the fermion propagators (Faulkner et al. Science 329, 1043, 2010)

Horizon geometry of the extremal Gravitational ‘mechanism’ for marginal black hole: ‘emergent’ AdS2 => (critical) Fermi-liquids: IR of boundary theory controlled G−1 = ω − v k − k − Σ k,ω by emergent temporal criticality F ( F ) ( )

€ Temporal scale invariance IR “lands” in probing fermion self energy! 2ν Σ"∝ω kF Fermi-surface “discovered” by matching UV-IR: like Mandelstam “fermion insertion” for Luttinger liquid! € 49 Holography and quantum matter

But first: crash course in holography

“Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy).

Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll).

Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

50 Phase diagram high Tc superconductors Mystery quantum ‘Stripy stuff’, spontaneous critical metal currents, phase fluctuations ..

The return of normalcy

+ + ΨBCS = Πk (uk + vkck↑c−k↓) vac. JZ, Science 315, 1372 (2007) € 51 “AdS-to-ARPES”: Fermi-liquid (?) emerging from a quantum critical state.

Cubrovic Schalm

52 The zero temperature extensive entropy ‘disaster’

The ‘extremal’ charged black hole with AdS2 horizon geometry has zero Hawking temperature but a finite horizon area. AdS-CFT

The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!) 53 Black hole hair can be fermionic! Schalm, Cubrovic, JZ (arXiv:1012.5681)

Stable Fermi liquid on the boundary!

AdS-CFT

‘Hydrogen atom’: quantum mechanical probability density ‘atmosphere’ of one fermion/surface area of black brane. 54 The Fermi-liquid VEV: Hair profile vs. statistics

Fermionic hair: the probability distribution along the radial direction of the AdS “hydrogen atom” wave function.

Position of the maximum determines the Fermi energy Fermionic hair: stability and equation of state.

Strongly renormalized EF Single Fermion spectral function: non Fermi-liquid Fermi surfaces have disappeared!

56 Holography and quantum matter

But first: crash course in holography

“Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev).

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like heavy fermions (Schalm, Cubrovic, Hartnoll).

Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

57 BCS theory: fermions turning into bosons Fermi-liquid + attractive interaction

Bardeen Cooper Schrieffer

Quasiparticles pair and Bose condense: Ground state D-wave SC: Dirac spectrum

+ + ΨBCS = Πk (uk + vkck↑c−k↓) vac.

€ 58 Superglue !

59 The zero temperature extensive entropy ‘disaster’

The ‘extremal’ charged black hole with AdS2 horizon geometry has zero Hawking temperature but a finite horizon area. AdS-CFT

The ‘seriously entangled’ quantum critical matter at zero temperature should have an extensive ground state entropy (?*##!!) 60 The holographic superconductor Hartnoll, Herzog, Horowitz, arXiv:0803.3295

Condensate (superconductor, … ) on the boundary!

AdS-CFT

‘Super radiance’: in the presence of matter the extremal BH is unstable => zero T entropy always (Scalar) matter ‘atmosphere’ avoided by low T order!!! 61 The Bose-Einstein hair cut.

Black hole scalar hair coding for the holographic superconductor

Scalar matter accumulates at the horizon Holographic superconductivity: stabilizing the fermions.

Fermion spectrum for scalar-hair back hole (Faulkner et al., 911.340; Chen et al., 0911.282):

Temperature dependence as expected for ‘quantum-critical’ superconductivity (She, JZ, 0905.1225)

‘BCS’ Gap in fermion Excessive temperature dependence spectrum !! ‘pacified’ ! 63 Hartnoll Herzog Horowitz Spielberg Thorne Fisk

Nature Nov 5 2009 Grigeria MacKenzie ThomsonRonning St Andrews Los Alamos

64 Fermionic quantum phase transitions in the heavy fermion metals

JZ, Science 319, 1205 (2008)

QP effective mass 1 ‘bad m* = E F actors’ * Coleman E F → 0 ⇒ m → ∞ Paschen et al., Nature (2004) Rutgers

€ Experimentalists: back to the entropic drawing board ..

Ruthenates: Lanthanides, actinides: St. Andrews Los Alamos

L in e of c ri ti ca l p o in ts

pq

Nailing down T=0 entropy hidden by last minute order: high precision entropy balance needed. Grigeria MacKenzie Thomson Ronning T c ΔC ΔSorder = ∫ dT 0 T 66

€ ?

Photoemission spectrum Further reading

AdS/CMT tutorials:

J. Mc Greevy, arXiv:0909.0518; S. Hartnoll, arXiv:0909.3553

AdS/CMT fermions: Hong Liu et al., arXiv:0903.2477,0907.2694,1003.0010; M. Cubrovic et al. Science 325,429 (2009), arXiv:1012.5681; T. Faulkner et al., Science 329, 1043 (2010). Condensed matter:

High Tc: J. Zaanen et al., Nature 430, 512, arXiv:1012.5461; 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 68 Quantum criticality or ‘conformal fields’

69 Fermi-liquid phenomenology

Bare single fermion propagator ‘enumerates the fixed point’: 1 Z G(ω,k)= 2 = ω − µ0 − k 2m − (Σʹ + iΣʹʹ) (ω − EF )− vR (k − kF )+K Σʹ ʹ ( ω,k) ImG(ω,k) = A ω,k = Spectral function: ( ) 2 2 2 ω + µ + (k − kF ) 2m + Σʹ ( ω,k) + Σʹ ʹ ( ω,k) 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: A(E F ,k) = Z δ(k − kF ) 2 - Analytical structure of the self-energy: Σʹ ʹ ω,k ∝ ω − E + ( ) ( F ) K ∂Σʹ ∂Σʹ Σʹ(ω,k)= Σʹ(EF ,kF )+ (ω − EF )+ (k − kF )+K ∂ω ∂k € ω=EF k =kF 2 - Temperature dependence: Σʹ ʹ ( E F ,kF ,T) ∝T +K € 70

€ Critical Fermi surfaces in heavy fermion systems

Blue = Fermi liquid Yellow= quantum critical regime

Antiferromagnetic order

FL Fermi surface Coexisting critical FL Fermi surface Fermi surfaces ? Marginal Fermi liquid phenomenology.

Fermi-gas interacting by second order perturbation theory with ‘singular heat bath’: ω ImP(q,ω) ∝−N(0) , for |ω |< T T ∝−N(0)sign(ω), for |ω |> T

Directly observed in e.g. Raman ?? 1 Single electron response €(photoemission): G(k,ω) = ω − vF (k − kF ) − Σ(k,ω) 2 ⎛ g ⎞ ⎡ π ⎤ Σ(k,ω) ∝⎜ ⎟ ⎢ω ln(max(|ω |,T)/ωc ) − i max(|ω |,T)⎥ ⎝ ωc ⎠ ⎣ 2 ⎦ € 1 Single particle life time ∝ max ( | ω |, T ) is coincident (?!) with the τ transport life time => linear resistivity. € 72 € Critical fermions at zero density: branchcut propagators

Two point Euclidean correlators: Ψ(τ,r ) = φ(τ,r ) φ(0,0)

r r Analytically continue to Minkowski time => susceptibilities χ t,r = Ψ iτ,r ( ) ( ) 1 1 1 € At criticality, conformal invariance: Ψ(τ) ∝ η ∝ Δ → χ(ω) ∝ Δ τ ωn (iω) 1 Lorentz invariance: € Scaling dimension set χ(ω,k) ∝ Δ −ω 2 + c 2k 2 by mass in AdS Dirac ( ) equation. €

€

73 AdS/CFT single fermion Spectral functions

Non-Fermi-liquid ν = 0.1

€

“Fermi-liquid” ν ≈1

0 €

74 Holographic Pauli-blocking: Lifshitz geometry.

2 2 2 2 2 dt dx + dy dz • Scaling metric: ds = − − z 2α z 2β z 2 Φ ∝ zκ , J 0 ∝ z 2γ , J 0 ∝ z 2γ +2δ , I ∝ z 2γ +δ • Scaling fields: − + • Scaling relations: α =1− 2m, β =1, κ =1, γ = 0, δ =1− 2m ‘Pseudogap’ fermions in high Tc superconductors

102 K

Tc = 82 K Gap stays open above Tc

But sharp quasiparticles disappear in incoherent 10 K ‘spectral smears’ in the metal Shen group, Nature 450, 81 (2007) 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.

2 ⎛ ⎞ 2ν −1 2 4 2 ∞ cTkF T − ⎜ ⎟ Fn (µ) ∂ Ωosc. πATckF πckF ebµ ⎝ µ ⎠ χ osc. = − 2 = 3 cos ∑e ∂B eB eB n= 0 77

€ Soaked in Entropy ….

Entropic catastrophe! d S = A + C T +L F = A T + L 78

€ Collective transport: fermion currents Hong Liu (MIT)

2ν Tedious one loop calculation, ‘accidental’ cancellations: ρFS ∝ Σ"1− fermion ∝T ‘Strange coincidence’ of one electron and transport lifetime of marginal fermi liquid finds gravitational explanation! € 79 ‘Shankar/Polchinski’ functional renormalization group

UV: weakly interacting Fermi gas

Integrate momentum shells: functions of running coupling constants

interaction All interactions (except marginal Hartree) irrelevant => Scaling Fermi sphere limit might be perfectly ideal Fermi-gas

80 The end of weak coupling

interaction Strong interactings:

Fermi gas as UV starting point does not make sense!

=> ‘emergent’ Fermi liquid fixed point remarkably resilient (e.g. 3He) Fermi sphere => Non Fermi-liquid/non ‘Hartree- Fock’ (BCS etc) states of fermion matter? 81 Numerics and fermionic

quantum criticality Jarrell

DCA results for Hubbard model at intermediate couplings (U = 0.75W): Non-fermi liquid ‘Mott fluid’ Quantum critical state, very unstable to d-wave superconductivity

Fermi-liquid at ‘high’ dopings 82 Graphene at the zero density Mott Transition

Herbut, Juricic, Vafek (arXiv:0904.1019): strongly interacting critical point at finite fermion coupling

83 Gravitationally coding the fermion propagators (Faulkner et al. Science Aug 27. 2010)

T=0 extremal black hole, near horizon geometry ‘emergent scale invariant’:

2ν k AdS2 ⊗ R2 ⇒ gk (ω) = c(k)ω Matching with the UV infalling Dirac waves:

GR (ω,k) = F0 (k) + F1(k)ω + F2 (k)gk (ω) Special momentum shell: | k | k € ≡ F h iγ 2ν G ( ,k) 1 ; ,k hg h e kF kF R ω = Σ(ω ) = kF (ω) = 2 ω € k − kF −ω /vF − Σ(ω,k) Space-like: IR-UV matching ‘organizes’ Fermi-surface. € Time-like: IR scale invariance picked up via AdS2 self energy € Miracle, this is like critical/marginal Fermi-liquids!! 84 AdS/CFT correspondence: String theory Magic! d-dim. gauge theory (d+1)-dim string theory / conformal field theory / gravity theory

boundary: strings d-dim space-time Black holes

Maldacena

gluons quarks Hawking radiation Witten, Gubser,Klebanov,Polyakov AdS/CFT: black holes and AdS-to-ARPES Holographic Entropic quantum critical planckian dissipation superconductivity singularities superconductivity

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