string theory and the mysterious quantum matter of condensed
TRANSCRIPT
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
Quantum critical cauliflower
Quantum critical cauliflower
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
€
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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Holography and quantum matter
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).
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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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:
€
ηs
=1
4πh
kB
Policastro-Son-Starinets (2002):
€
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!
€
4πkBηhs
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Quantum critical hydrodynamics:Planckian dissipation & viscosity
Planckian dissipation:
€
h
kBT
Viscosity, entropy density:
Planckian viscosity:
€
η = ε + p( )τ, s =ε + p
T⇒
ηs
= Tτ
€
τ = τ h ≈h
kBT
€
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 ..
€
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!
€
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:
€
⇒ [ h
kBTσ1
] = const.(1+ A2[ hωkBT
]2)
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Divine resistivity = PlanckianDissipation!
€
ρ ∝1τ h
∝ kBT
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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 heavy fermions (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, perfectfluids and the linear resistivity (Son, Policastro, …, Sachdev).
But first: crash course in holography
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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
€
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:
€
G−1 =ω − vF k − kF( ) − Σ k,ω( )
€
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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Holography and quantum matter
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 heavy fermions (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, perfectfluids and the linear resistivity (Son, Policastro, …, Sachdev).
But first: crash course in holography
51
Phase diagram high Tcsuperconductors
JZ, Science 315,1372 (2007)
Mystery quantumcritical metal‘Stripy stuff’, spontaneous
currents, phase fluctuations ..
€
ΨBCS =Πk uk + vkck↑+ c−k↓
+( ) 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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Holography and quantum matter
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 heavy fermions (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, perfectfluids and the linear resistivity (Son, Policastro, …, Sachdev).
But first: crash course in holography
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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
€
ΨBCS =Πk uk + vkck↑+ c−k↓
+( ) 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)
€
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.
€
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
€
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:
€
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
,,
€
ʹ′ ʹ′ Σ ω,k( )∝ ω − EF( )2+K
- Temperature dependence:
€
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’:
€
ImP(q,ω)∝−N(0)ωT
, for |ω |< T
∝−N(0)sign ω( ), for |ω |> T
Directly observed in e.g. Raman ??
€
G(k,ω) =1
ω − vF k − kF( ) − Σ(k,ω)
€
Σ(k,ω)∝ gωc
⎛
⎝ ⎜
⎞
⎠ ⎟
2
ω ln max |ω |,T( ) /ωc( ) − i π2
max |ω |,T( )⎡
⎣ ⎢ ⎤
⎦ ⎥
€
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
€
Ψ τ,r r ( ) = φ τ,r r ( ) φ(0,0)
€
χ t,r r ( ) = Ψ iτ,r r ( )
At criticality, conformal invariance:
€
Ψ τ( )∝ 1τη
∝1ωn
Δ → χ(ω)∝ 1iω( )Δ
Lorentz invariance:
€
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
€
ν = 0.1
€
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.
€
χosc. = −∂ 2Ωosc.
∂B2 =πATckF
4
eB3 cosπckF2
eBe−
cTkF2
ebµTµ
⎛
⎝ ⎜
⎞
⎠ ⎟
2ν −1
Fn µ( )
n= 0
∞
∑
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Soaked in Entropy ….
€
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)
€
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)
€
GR ω,k( ) = F0 k( ) + F1 k( )ω + F2(k)gk ω( )
€
| k |≡ kF
€
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’:
€
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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