Post on 11-Feb-2022
transcript
String theory and theString theory and themysterious quantum matter ofmysterious quantum matter of
condensed matter physics.condensed matter physics.Jan Zaanen
1
2
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
3
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
6
Hairy Black holes …
7
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
8
General relativity “=“ quantumfield theory
Gravity Quantum fields
Maldacena 1997
=
9
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
11
1
1
1
1
1
1
1
11
1
11
1
00
0
0 01
0 0
10
00
0
0 00
01
1
1
1
10
0 0
1
1
0 1
11
0
1001
101
1
1
1
1 0
0
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)!!!
21
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!
22
Black hole hair codes thequantum matter
“Hairy black holes”code for (un)stablestates of quantummatter emerging fromthe quantum criticalstuff.
23
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
25
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
26
27
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
28
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)
30
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
32
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
33
Twenty five years ago …Mueller Bednorz
Ceramic CuO’s,likeYBa2Cu3O7
Superconductivityjumps to ‘high’temperatures
34
Graveyard of Theories
Schrieffer
Anderson
Mueller
Bednorz
Laughlin
Abrikosov Leggett
Wilczek
Mott
Ginzburg
De Gennes
YangLee
35
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
36
Quantum Phase transitions
Quantum scale invariance emerges naturally at a zero temperaturecontinuous phase transition driven by quantum fluctuations:
JZ, Science 319, 1205 (2008)
37
A universal phase diagram
Quantumcritical
Heavy fermionsHigh Tcsuperconductors
Ironsuperconductors (?)
Quantumcritical
38
Divine resistivity
39
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)
40
Divine resistivity = PlanckianDissipation!
€
ρ ∝1τ h
∝ kBT
41
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
42
Holographic quantum criticalfermion state
Liu McGreevy
43
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.
44
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
45
ARPES: Observing Fermi liquids
‘MDC’ at EF in conventional2D metal (NbSe2)
Fermi-liquids: sharp Quasiparticle ‘poles’
46
Cuprates: “Marginal” or “Critical”Fermi liquids
Fermi ‘arcs’ (underdoped)closing to Fermi-surfaces(optimally-, overdoped).
EDC lineshape: ‘branch cut’ (conformal),width propotional to energy
47
Breaking fermionic criticalitywith a chemical potential
‘Dirac waves’
Electrical monopole
k
E
€
µ
€
µ
Fermi-surface??
48
AdS/ARPES for the Reissner-Nordstrom non-Fermi liquids
Critical FL Marginal FL Non Landau FL
Fermi surfaces but no quasiparticles!
49
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)
50
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
52
“AdS-to-ARPES”: Fermi-liquid (?)emerging from a quantum critical state.
SchalmCubrovic
53
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 (?*##!!)
54
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
56
Fermionic hair: stability andequation of state.
Strongly renormalized EF Single Fermion spectral function:non Fermi-liquid Fermi surfaceshave disappeared!
57
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
58
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
59
Superglue !
60
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 (?*##!!)
61
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
63
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’ !
64
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
66
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
68
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
69
Quantum criticality or ‘conformalfields’
70
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 ?
72
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.
73
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.
74
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
76
‘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)
77
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
∞
∑
78
Soaked in Entropy ….
€
S = A + C T d +L
F = A T +L
Entropic catastrophe!
79
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!
80
‘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
81
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?
82
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
83
Graphene at the zerodensity Mott Transition
Herbut, Juricic, Vafek (arXiv:0904.1019):strongly interacting critical point atfinite fermion coupling
84
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
86
Entropicsingularities
AdS/CFT: black holes andplanckian dissipation
AdS-to-ARPES Holographicsuperconductivity
quantum criticalsuperconductivity
87
Empty
88
Empty
89
Empty