String theory and the String theory and the mysterious quantum matter of mysterious quantum matter of condensed matter physics. condensed matter 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 Magneto-optics High-resolution ARPES 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 Quantum fields Gravity 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 Another word for: universe … Quantum criticality! 9
Holography with lasers Encoded on a two Three dimensional image dimensional photographic plate
Gravity - quantum field holography Quantum fields in flat space Einstein world “AdS” = “CFT”= quantum critical Anti de Sitter universe
Hawking radiation
‘t Hooft’s holographic principle Hawking Temperature: g = acceleration at horizon BH entropy: 1 0 0 1 0 0 1 0 1 0 1 A = area of horizon 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 1 Number of degrees of freedom (field 1 1 1 0 1 1 0 1 theory) scales with the area and not 1 0 0 0 0 0 1 1 1 1 with the volume (gravity) 1 0 1 1 0 0 1
The bulk: Anti-de Sitter space Extra radial dimension of the bulk <=> scaling “dimension” in the field theory IR UV 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: d-dim space-time strings 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 ( ) = S AdS φ ( ) φ x 0 → 0 = J W CFT J 2 N = R 4 g YM α = g s 2 g YM 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 Quantum-critical fields on the boundary: GR in Anti de Sitter space - 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 ( ) η = σ abs 0 gravitons by black hole 16 π G = area of horizon (GR theorems) Entropy density s: Bekenstein-Hawking BH entropy = area of horizon h = 1 η Universal viscosity-entropy ratio for CFT’s with gravitational dual limited in large N by: 4 π s k B 28
AdS/CFT viscosity Kovtun-Son-Starinets (2005) 4 π k B η h s
The quark-gluon plasma Relativistic Heavy Ion Collider Quark-gluon ‘fireball’ 30
The tiny viscosity of the Quark- Gluon plasma 4 π k B η h s 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 h state the time it takes to convert work in heat (relaxation time) has to be k B T h τ = τ h ≈ Sachdev, 1992 k B T ) τ , s = ε + p ⇒ η ( η = ε + p s = T τ Viscosity, entropy density: T s ≈ h η Planckian viscosity: k B 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 Anderson Leggett De Gennes Bednorz Lee Wilczek Ginzburg Yang 34
Phase diagram high Tc superconductors Mystery quantum critical ‘Stripy stuff’, spontaneous currents, phase fluctuations .. metal The return of normalcy ( ) vac . + c − k ↓ + Ψ BCS = Π k u k + v k c k ↑ 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 τ r ω pr τ r = A h σ 1 ( ω , T ) = 1 2 , 1 + ω 2 τ r 4 π k B T h ] = const .(1 + A 2 [ h ω k B T ] 2 ) [ ⇒ k B T σ 1 A= 0.7 : the normal state of optimallly doped cuprates is a Planckian dissipator ! 39
Divine resistivity = Planckian Dissipation! ρ ∝ 1 ∝ k B T τ 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 Pauli exclusion principle: every energy state occupied by one electron Unreasonable: electrons strongly interact !! Fermi momenta k=1/wavelength Landau’s Fermi-liquid: the highly collective low energy quantum excitations are like electrons that do not interact. Fermi surface of copper 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 44 k=1/wavelength
ARPES: Observing Fermi liquids ‘MDC’ at E F in conventional Fermi-liquids: sharp Quasiparticle ‘poles’ 2D metal (NbSe 2 ) 45
Cuprates: “Marginal” or “Critical” Fermi liquids Fermi ‘arcs’ (underdoped) EDC lineshape: ‘branch cut’ (conformal), closing to Fermi-surfaces width propotional to energy (optimally-, overdoped). 46
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