String theory and the mysterious quantum matter of condensed

String theory and theString theory and themysterious quantum matter ofmysterious quantum matter of

condensed matter physics.condensed matter physics.Jan Zaanen

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String theory: what is it really goodfor?

- Hadron (nuclear) physics: quark-gluon plasma in RIHC.

- Quantum matter: quantum criticality in heavy fermionsystems, high Tc superconductors, …

Started in 2001, got on steam in 2007.

Son Hartnoll Herzog Kovtun McGreevy Liu Schalm

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Quantum critical matter

Quantumcritical

Heavy fermionsHigh Tcsuperconductors

Ironsuperconductors (?)

Quark gluon plasma

Quantumcritical

High-Tc Has Changed Landscape of Condensed Matter PhysicsHigh-resolution ARPES

Spin-polarized Neutron

Magneto-optics

STM

Transport-Nernst effect

High TcSuperconductivity

Angle-resolved MR/Heat CapacityInelastic X-Ray Scattering

?

Photoemissionspectrum

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Hairy Black holes …

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Holography and quantum matter

Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’snormal 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 forsuperconductivity at a high temperature (Hartnoll, Herzog,Horowitz) .

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

But first: crash course in holography

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General relativity “=“ quantumfield theory

Gravity Quantum fields

Maldacena 1997

=

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Anti de Sitter-conformal quantumfield theory correspondence

AdS geometry(“near” the boundary)

Conformal quantum fieldtheory (at ‘high’ energies)

Another word for:Quantum criticality!

Not like ouruniverse …

Holography with lasers

Three dimensional image Encoded on a twodimensionalphotographic plate

Gravity - quantum fieldholography

Einstein world “AdS” =Anti de Sitter universe

Quantum fields in flat space“CFT”= quantum critical

Hawking radiation

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Hawking Temperature:

g = acceleration at horizon

A = area of horizon

‘t Hooft’s holographic principle

BH entropy:

Number of degrees of freedom (fieldtheory) scales with the area and notwith the volume (gravity)

The bulk: Anti-de Sitter space

Extra radial dimensionof the bulk <=> scaling“dimension” in the fieldtheory

Bulk AdS geometry =scale invariance ofthe field theory

UVIR

Fractal Cauliflower (romanesco)

Quantum critical cauliflower

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-hardproblem (Troyer, Wiese)!!!

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Renormalization group forquantum critical matter

Wilson-Fisher RG:based on Boltzmannianstatistical physics

boundary: d-dim space-time

Hawking radiationgluons

Black holesstrings

quarks

The Magic of AdS/CFT!

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Black hole hair codes thequantum matter

“Hairy black holes”code for (un)stablestates of quantummatter emerging fromthe quantum criticalstuff.

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Quantum critical dynamics:classical waves in AdS

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WCFT J( ) = SAdS φ( )φx0 →0= J

gYM2 N =

R4

αgYM

2 = gs

E-fieldtransverse E-field <=> 3d electric fieldradial E-field <=> 3d charge density

B-fieldradial B-field <=> 3d magnetic fieldtransverse B-field <=> 3d current density

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

temporal metric perturb.transverse gradient <=> temperature gradientradial gradient <=> heat flow

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

The AdS/CFT dictionary

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Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normalstate (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).


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


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The Schwarzschild Black Hole isthe heater

GR in Anti de Sitter space Quantum-critical fields on the boundary:

Black holetemperatureentropy

- at the Hawking temperature- entropy = black hole entropy

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Dissipation = absorption ofclassical waves by Black hole!

Viscosity: absorption cross section ofgravitons by black hole

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

η =σ abs 0( )16πG

= area of horizon (GR theorems)

Universal viscosity-entropy ratio for CFT’swith gravitational dual limited in large N by:

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ηs

=1

4πh

kB

Policastro-Son-Starinets (2002):

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4πkBηhs

AdS/CFT viscosityKovtun-Son-Starinets (2005)

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The quark-gluon plasma

Relativistic Heavy Ion Collider Quark-gluon ‘fireball’

The tiny viscosity of the Quark-Gluon plasma

QG plasma:within 20% ofthe AdS/CFTviscosity!

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4πkBηhs

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Quantum critical hydrodynamics:Planckian dissipation & viscosity

Planckian dissipation:

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h

kBT

Viscosity, entropy density:

Planckian viscosity:

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η = ε + p( )τ, s =ε + p

T⇒

ηs

= Tτ

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τ = τ h ≈h

kBT

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ηs≈

h

kB

In a finite temperature quantum criticalstate the time it takes to convert work inheat (relaxation time) has to be €

1ω

Sachdev,1992

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Twenty five years ago …Mueller Bednorz

Ceramic CuO’s,likeYBa2Cu3O7

Superconductivityjumps to ‘high’temperatures

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Graveyard of Theories

Schrieffer

Anderson

Mueller

Bednorz

Laughlin

Abrikosov Leggett

Wilczek

Mott

Ginzburg

De Gennes

YangLee

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Phase diagram high Tcsuperconductors

JZ, Science 315,1372 (2007)

Mysteryquantum criticalmetal

‘Stripy stuff’, spontaneouscurrents, phase fluctuations ..

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ΨBCS =Πk uk + vkck↑+ c−k↓

+( ) vac.

The return ofnormalcy

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Quantum Phase transitions

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

JZ, Science 319, 1205 (2008)

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A universal phase diagram

Quantumcritical

Heavy fermionsHigh Tcsuperconductors

Ironsuperconductors (?)

Quantumcritical

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Divine resistivity

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Critical Cuprates are PlanckianDissipators

A= 0.7: the normal state of optimallly doped cuprates is aPlanckian dissipator!

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σ1(ω,T) =1

4πω pr

2 τ r

1+ω 2τ r2 , τ r = A h

kBT

van der Marel, JZ, … Nature 2003:

Optical conductivity QC cuprates

Frequency less than temperature:

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⇒ [ h

kBTσ1

] = const.(1+ A2[ hωkBT

]2)

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Divine resistivity = PlanckianDissipation!

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ρ ∝1τ h

∝ kBT

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Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’snormal 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).


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


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Holographic quantum criticalfermion state

Liu McGreevy

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The quantum in the kitchen:Landau’s miracle

Kinetic energy

k=1/wavelength

Electrons are waves

Pauli exclusion principle: everystate occupied by one electron

Fermi momenta

Fermienergy

Fermi surface of copper

Unreasonable: electrons stronglyinteract !!

Landau’s Fermi-liquid: thehighly collective low energyquantum excitations are likeelectrons that do not interact.

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Watching electrons:photoemission

Kinetic energy

k=1/wavelengthFermi momenta

Fermienergy

Fermi surface of copper

Electron spectral function: probability tocreate or annihilate an electron at agiven momentum and energy.

k=1/wavelength

Fermienergy

energy

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ARPES: Observing Fermi liquids

‘MDC’ at EF in conventional2D metal (NbSe2)

Fermi-liquids: sharp Quasiparticle ‘poles’

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Cuprates: “Marginal” or “Critical”Fermi liquids

Fermi ‘arcs’ (underdoped)closing to Fermi-surfaces(optimally-, overdoped).

EDC lineshape: ‘branch cut’ (conformal),width propotional to energy

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Breaking fermionic criticalitywith a chemical potential

‘Dirac waves’

Electrical monopole

k

E

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µ

€

µ

Fermi-surface??

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AdS/ARPES for the Reissner-Nordstrom non-Fermi liquids

Critical FL Marginal FL Non Landau FL

Fermi surfaces but no quasiparticles!

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Horizon geometry of the extremalblack hole: ‘emergent’ AdS2 =>IR of boundary theory controlledby emergent temporal criticality

Gravitational ‘mechanism’ for marginal(critical) Fermi-liquids:

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G−1 =ω − vF k − kF( ) − Σ k,ω( )

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Σ"∝ω 2ν kF

Fermi-surface “discovered” by matchingUV-IR: like Mandelstam “fermioninsertion” for Luttinger liquid!

Temporal scale invariance IR “lands” inprobing fermion self energy!

Gravitationally coding the fermionpropagators (Faulkner et al. Science 329, 1043, 2010)

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Phase diagram high Tcsuperconductors

JZ, Science 315,1372 (2007)

Mystery quantumcritical metal‘Stripy stuff’, spontaneous

currents, phase fluctuations ..

€


+( ) vac.

The returnof normalcy

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“AdS-to-ARPES”: Fermi-liquid (?)emerging from a quantum critical state.

SchalmCubrovic

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The zero temperature extensiveentropy ‘disaster’

AdS-CFT

The ‘extremal’ charged blackhole with AdS2 horizongeometry has zero Hawkingtemperature but a finitehorizon area.

The ‘seriously entangled’quantum critical matter atzero temperature should havean extensive ground stateentropy (?*##!!)

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Black hole hair can be fermionic!Schalm, Cubrovic, JZ (arXiv:1012.5681)

‘Hydrogen atom’: quantum mechanicalprobability density ‘atmosphere’ of onefermion/surface area of black brane.

AdS-CFT

Stable Fermi liquid on theboundary!

The Fermi-liquid VEV:Hair profile vs. statistics

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

Position of the maximumdetermines the Fermi energy

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Fermionic hair: stability andequation of state.

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

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BCS theory: fermions turning intobosons

Fermi-liquid + attractive interaction

Bardeen Cooper Schrieffer

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

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+( ) vac.

Ground state

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Superglue !

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The zero temperature extensiveentropy ‘disaster’

AdS-CFT

The ‘extremal’ charged blackhole with AdS2 horizongeometry has zero Hawkingtemperature but a finitehorizon area.

The ‘seriously entangled’quantum critical matter atzero temperature should havean extensive ground stateentropy (?*##!!)

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The holographic superconductorHartnoll, Herzog, Horowitz, arXiv:0803.3295

(Scalar) matter ‘atmosphere’

AdS-CFT

Condensate (superconductor,… ) on the boundary!

‘Super radiance’: in thepresence of matter theextremal BH is unstable =>zero T entropy alwaysavoided by low T order!!!

The Bose-Einstein hair cut.

Black hole scalar hair coding for the holographic superconductor

Scalar matter accumulatesat the horizon

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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 fermionspectrum !!

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

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Spielberg ThorneHartnoll Herzog Horowitz

Fisk

ThomsonRonningMacKenzieGrigeria

Los AlamosSt Andrews

Nature Nov 5 2009

Fermionic quantum phase transitionsin the heavy fermion metals

Paschen et al., Nature (2004)

JZ, Science 319, 1205(2008)

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m* =1

EF

EF → 0⇒ m* →∞

QP effective mass‘badactors’

ColemanRutgers

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Experimentalists: back to theentropic drawing board ..

Grigeria MacKenzie Thomson Ronning

Nailing down T=0 entropy hiddenby last minute order: highprecision entropy balance needed.

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ΔSorder =ΔCT0

Tc

∫ dT

Lanthanides, actinides:Los Alamos

Ruthenates:St. Andrews

Line of critical s

point

pq

?

Photoemissionspectrum

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Further readingAdS/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. Varmaet al., Phys. Rep. 361, 267417

Heavy Fermions: J. Zaanen, Science 319, 1205; von Loehneisen et al, Rev.Mod. Phys. 79, 1015

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Quantum criticality or ‘conformalfields’

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Fermi-liquidphenomenology

Bare single fermion propagator ‘enumerates the fixed point’:

Spectral function:

( )( ) ( ) ( ) K+−−−

=Σ ʹ′ʹ′+Σʹ′−−−

=FRF kkvE

Zimk

kGωµω

ω21, 2

0

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ImG(ω,k) = A ω,k( ) =ʹ′ ʹ′ Σ ω,k( )

ω + µ + k − kF( )2 2m + ʹ′ Σ ω,k( )2

+ ʹ′ ʹ′ Σ ω,k( )2

The Fermi liquid ‘lawyer list’:

- At T= 0 the spectral weight is zero at the Fermi-energy except for thequasiparticle peak at the Fermi surface:

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A EF ,k( ) = Z δ k − kF( )

- Analytical structure of the self-energy:( ) ( ) ( ) ( ) K+−

∂

Σʹ′∂+−

∂

Σʹ′∂+Σʹ′=Σʹ′

==F

kkF

EFF kk

kEkEk

FF

ωω

ωω

,,

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ʹ′ ʹ′ Σ ω,k( )∝ ω − EF( )2+K

- Temperature dependence:

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ʹ′ ʹ′ Σ EF ,kF ,T( )∝T 2 +K

Critical Fermi surfaces in heavyfermion systems

Blue = Fermi liquid

Yellow= quantumcritical regime

Antiferromagneticorder

FL Fermi surface FL Fermi surfaceCoexisting criticalFermi surfaces ?

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Marginal Fermi liquidphenomenology.

Fermi-gas interacting by second order perturbation theory with ‘singular heat bath’:

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ImP(q,ω)∝−N(0)ωT

, for |ω |< T

∝−N(0)sign ω( ), for |ω |> T

Directly observed in e.g. Raman ??

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G(k,ω) =1

ω − vF k − kF( ) − Σ(k,ω)

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Σ(k,ω)∝ gωc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

ω ln max |ω |,T( ) /ωc( ) − i π2

max |ω |,T( )⎡

⎣ ⎢ ⎤

⎦ ⎥

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1τ∝max |ω |,T( )

Single electron response (photoemission):

Single particle life time is coincident (?!) with thetransport life time => linear resistivity.

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Critical fermions at zero density:branchcut propagators

Two point Euclidean correlators:

Analytically continue to Minkowski time => susceptibilities

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Ψ τ,r r ( ) = φ τ,r r ( ) φ(0,0)

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χ t,r r ( ) = Ψ iτ,r r ( )

At criticality, conformal invariance:

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Ψ τ( )∝ 1τη

∝1ωn

Δ → χ(ω)∝ 1iω( )Δ

Lorentz invariance:

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χ ω,k( )∝ 1

−ω 2 + c 2k 2( )Δ

Scaling dimension setby mass in AdS Diracequation.

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AdS/CFT single fermion Spectralfunctions

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ν = 0.1

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ν ≈1

Non-Fermi-liquid

“Fermi-liquid”

0

• Scaling metric:

• Scaling fields:• Scaling relations:

Holographic Pauli-blocking:Lifshitz geometry.

δγδγγκ +++− ∝∝∝∝Φ 222020 ,,, zIzJzJz

mm 21,0,1,1,21 −====−= δγκβα

2

2

2

22

2

22

zdz

zdydx

zdtds −

+−=

βα

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‘Pseudogap’ fermions in high Tcsuperconductors

10 K

Tc = 82 K

102 K

Gap stays open above Tc

But sharp quasiparticlesdisappear in incoherent‘spectral smears’ in the metal

Shen group, Nature 450, 81 (2007)

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Thermodynamics: where are thefermions?

Hartnoll et al.: arXiv:0908.2657,0912.0008Large N limit: thermodynamics entirely determined byAdS geometry.

Fermi surface dependent thermodynamics, e.g. Haas vanAlphen oscillations?

Leading 1/N corrections: “Fermionic one-loopdark energy”

Quantum corrections: one loop using Dirac quasinormal modes:‘generalized Lifshitz-Kosevich formula’ for HvA oscillations.

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χosc. = −∂ 2Ωosc.

∂B2 =πATckF

4

eB3 cosπckF2

eBe−

cTkF2

ebµTµ

⎛

⎝ ⎜

⎞

⎠ ⎟

2ν −1

Fn µ( )

n= 0

∞

∑

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Soaked in Entropy ….

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S = A + C T d +L

F = A T +L

Entropic catastrophe!

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Collective transport: fermioncurrents

Tedious one loop calculation, ‘accidental’ cancellations:

Hong Liu (MIT)

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ρFS ∝Σ"1− fermion∝T 2ν

‘Strange coincidence’ of one electron and transport lifetime of marginal fermiliquid finds gravitational explanation!

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‘Shankar/Polchinski’ functionalrenormalization group

interaction

Fermi sphere

UV: weakly interacting Fermi gas

Integrate momentum shells:functions of running couplingconstants

All interactions (except marginalHartree) irrelevant => Scalinglimit might be perfectly idealFermi-gas

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The end of weak coupling

interaction

Fermi sphere

Strong interactings:

Fermi gas as UV starting pointdoes not make sense!

=> ‘emergent’ Fermi liquid fixedpoint remarkably resilient (e.g. 3He)

=> Non Fermi-liquid/non ‘Hartree-Fock’ (BCS etc) states of fermionmatter?

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Numerics and fermionicquantum criticality Jarrell

DCA results for Hubbard model at intermediate couplings (U = 0.75W):Non-fermi liquid ‘Mott fluid’

Fermi-liquid at ‘high’ dopings

Quantum critical state, very unstable tod-wave superconductivity

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Graphene at the zerodensity Mott Transition

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

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Gravitationally coding the fermionpropagators (Faulkner et al. Science Aug 27. 2010)

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GR ω,k( ) = F0 k( ) + F1 k( )ω + F2(k)gk ω( )

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| k |≡ kF

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GR (ω,k) =h1

k − kF −ω /vF − Σ ω,k( ); Σ ω,k( ) = hgkF

ω( ) = h2eiγ kFω 2ν kF

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

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AdS2 ⊗ R2 ⇒ gk ω( ) = c k( )ω 2ν k

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

boundary: d-dim space-time

Hawking radiationgluons

Black holesstrings

quarks

AdS/CFT correspondence: String theory Magic!

d-dim. gauge theory (d+1)-dim string theory/ conformal field theory / gravity theory

Maldacena

Witten, Gubser,Klebanov,Polyakov

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Entropicsingularities

AdS/CFT: black holes andplanckian dissipation

AdS-to-ARPES Holographicsuperconductivity

quantum criticalsuperconductivity

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String theory and the mysterious quantum matter of condensed

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