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Planckian dissipation
Jan Zaanen
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The dissipative world of apes …
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The Gross list: the 14 Big Questions
2004
1. The origin of the universe?2. What is dark matter?
11. What is space-time?
0. The origin of temperature, dissipation and probability?
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QuickTime™ and a decompressor
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Subir Sachdev (Harvard)
Sudip Chakravarty (UCLA)
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Plan
1. The dissipative world of apes ….
2. The high Tc superconductivity saga: the past.
3. Viscosity, dissipation and black holes: the AdS/CFT magic
5. Conclusions
4. The high Tc superconductivity saga: the future is critical.
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The second law ….
Dissipation: useful work has to turn into worthless heat …
dSCl
dt 0
Let’s go quantum:
SvN Tr ln
ij ij
eE i
eE i
i
SvN SCl
it ,H
dSvN
dt 0
Von Neumann
Thermal density matrix:
Unitary time evolution:
The second law is beyond quantum physics !!
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Unitarity versus general covariance
Penrose Susskind ‘t Hooft
See also: Van Wezel, Oosterkamp, JZ, cond-mat/0706.3976
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Quantum dissipation in equilibrium worlds
"(k ,)
1
21 e S
k ,
Fluctuation-dissipation theorem:
Green-Kubo-Matsubara
Thermal quantum field theory = Euclidean path integrals
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Wick rotation: time versus temperature
Two point Euclidean correlators:
Analytically continue to ‘our’
Minkowski time => susceptibilities = observables
,r ,
r (0,0)
t,r i,
r
m
m
Measurement time short compared to : „coherent‟
= unitary dynamics
kBT
Measurement time long compared to : „overdamped‟ =
dissipative dynamics
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QuickTime™ and a decompressor
are needed to see this picture.
Subir Sachdev (Harvard)
Sudip Chakravarty (UCLA)
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1
1
1
1
1
1
1
1
00
0 01
00
0
0 00
1
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0
0
1
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0
Hawking Temperature & Entropy
g = acceleration at horizon
A = area of horizon
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Minkowski versus Euclidean black holes
Gibbons-Hawking
Schwarzschild in real time
horizon
Radial distance
Imaginary time
R
kBTHawking
Schwarzschild in imaginary time
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Planck scales
Phenomenon: gravity, matter warps space-time
Characteristic dimensions: Newton’s constant G,
light velocity c.
Planck’s constant carries dimension energy-seconds:
Einstein comes to an end at the Planck scale
Phenomenon: dissipation, work turns irreversibly into heat, takes a characteristic (relaxation) time
Characteristic dimension: temperature
Given hbar, the shortest possible relaxation time is
‘Planckian dissipation’: requires that the quantum
dynamics is scale invariant (quantum critical)!!!!
G G c 5 1042 sec
kBT
Planck
kBT
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Plan
1. The dissipative world of apes ….
2. The high Tc superconductivity saga: the past.
3. Viscosity, dissipation and black holes: the AdS/CFT magic
5. Conclusions
4. The high Tc superconductivity saga: the future is critical.
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The quantum in the kitchen: Landau’s miracle
Kinetic energy
k=1/wavelength
Electrons are waves
Pauli exclusion principle: every state occupied by one electron
Fermi momenta
Fermi energy
Fermi surface of copper
Unreasonable: electrons strongly interact !!
Landau’s Fermi-liquid: the
highly collective low energy quantum excitations are like electrons that do not interact.
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BCS theory: fermions turning into bosons
Fermi-liquid + attractive interaction
Bardeen
Cooper Schrieffer
Quasiparticles pair and Bose condense: D-wave SC: Dirac spectrum
BCS k uk vkck
ck
vac.
Ground state
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Twenty two years ago …
Mueller Bednorz
Ceramic CuO’s,
likeYBa2Cu3O7
Superconductivity jumps to „high‟
temperatures
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Graveyard of Theories
Schrieffer
Anderson
Mueller
Bednorz
Laughlin
Abrikosov
Leggett
Wilczek
Mott
Ginzburg
De Gennes
Yang
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Phase diagram high Tc superconductors
JZ, Science 315, 1372 (2007)
Mystery quantum critical metal‘Stripy stuff’, spontaneous
currents, phase fluctuations ..
BCS k uk vkck
ck
vac.
The return of normalcy
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Divine resistivity
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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-hard
problem (Troyer, Wiese)!!!
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Planck scales
Phenomenon: gravity, matter warps space-time
Characteristic dimensions: Newton’s constant G,
light velocity c.
Planck’s constant carries dimension energy-seconds:
Einstein comes to an end at the Planck scale
Phenomenon: dissipation, work turns irreversibly into heat, takes a characteristic (relaxation) time
Characteristic dimension: temperature
Given hbar, the shortest possible relaxation time is
‘Planckian dissipation’: requires that the quantum
dynamics is scale invariant (quantum critical)!!!!
G G c 5 1042 sec
kBT
Planck
kBT
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Quantum criticality or ‘conformal fields’
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Fractal Cauliflower (romanesco)
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The quantum critical response
1
kBT ??
1
T 2
1
1 i
const.
kBT, const.O(1)Planckian
dissipation:
Sachdev
kBT
Scaling form dynamical susceptibility:
1
T 2(
kBT)
Quantum critical regime
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Dissipation in scale-full quantum systems
Example: dissipation in the Fermi-liquid (electron-electron scattering)
nee
2
m inel.
inel. EF
(kBT )2
EF
kBT
EF /kBTIt takes a time longer by
than the Planck time
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Critical Cuprates are Planckian Dissipators
A= 0.7: the normal state of optimallly doped
cuprates is a Planckian dissipator!
1(,T) 1
4
pr
2 r
1 2 r
2, r A
kBT
van der Marel, JZ, … Nature 2004:Optical conductivity QC cuprates
Frequency less than temperature:
[
kBT1] const.(1 A2[
kBT]2)
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Divine resistivity
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‘Real’ hydrodynamics:
Planckian viscosity
1
Planckian dissipation:
kBT
Viscosity, entropy density:
Planckian viscosity:
p
s p
T
s T
kBT
s
kB
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Plan
1. The dissipative world of apes ….
2. The high Tc superconductivity saga: the past.
3. Viscosity, dissipation and black holes: the AdS/CFT magic
5. Conclusions
4. The high Tc superconductivity saga: the future is critical.
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boundary:
d-dim space-time
Hawking radiationgluons
Black holes
strings
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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s
The bulk: Anti-de Sitter space
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The boundary: conformal fields or quantum
criticality!
GR in Anti de Sitter space
Quantum-critical fields on the boundary
Planckian dissipation!
Black hole temperature entropy
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Quantum critical dynamics: classical waves in AdS
WCFT J SAdS x0
0 J
gYM
2 N R4
gYM
2 gs
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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
SUSY Einstein-Maxwell in AdS <==> SUSY Yang-Mills CFT
The AdS/CFT dictionary
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Dissipation = absorption of classical waves by Black
hole!
Viscosity: absorption cross section of gravitons by black hole
Entropy density s: Bekenstein-Hawking BH entropy = area of horizon
abs 0 16G
= area of horizon (GR theorems)
Universal viscosity-entropy ratio for CFT‟s with gravitational dual limited in
large N by:
s1
4
kB
Hartnoll-Son-Starinets (2002):
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‘Real’ hydrodynamics:
Planckian viscosity
1
Planckian dissipation:
kBT
Viscosity, entropy density:
Planckian viscosity:
p
s p
T
s T
kBT
s
kB
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4kB
s
AdS/CFT viscosity Kovtun-Son-Starinets (2005)
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The quark-gluon plasma
Relativistic Heavy Ion Collider
Quark-gluon ‘fireball’
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The tiny viscosity of the Quark-Gluon plasma
QG plasma: within 20% of the AdS/CFT viscosity!
4kB
s
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S=1/2 Fermi gas at a Feshbach resonance
Very cold fermionic atoms
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RG fixed point described by a “non-relativistic”
CFT: special gravitational AdS dual
Quantum criticality at the Feshbach resonance
See arXiv:0804.3972,0804.4053,0806.2867,0806.3244,0807.1100,0807.1111
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T. Schafer, Phys. Rev. A 76, 063618 (2007); A. Turlapov et. al., J. Low Temp. Physics 150, 567 (2008)
s1
4
kBQuark Gluon plasma
Ultracold 6-Li gas at Feshbach resonance
Planckian dissipation in the atom trap
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Plan
1. The dissipative world of apes ….
2. The high Tc superconductivity saga: the past.
3. Viscosity, dissipation and black holes: the AdS/CFT magic
5. Conclusions
4. The high Tc superconductivity saga: the future is critical.
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48
Thermo-electric transport at the insulator-superfluid QPT.
Hartnoll, Kovtun, Müller, Sachdev,
Phys. Rev. B 76 144502 (2007)
E.g. Nernst effect:
AdS: graviton-photon cross correlators in the presence of a dyonic black hole!
Stripy cuprates: Ong group
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49
Why Tc is high …JZ, Nature 430, 512(2004)
Fermionic quantum critical state
BCS type transition: pairs form at Tc
2 bosone1/
2 3.5kBTc
Need the Fermi energy!
But BCS wisdoms like:
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The hairy black holeHartnoll, Herzog, Horowitz, arXiv:0803.3295
Condensate „atmosphere‟
2 0
kBTc
8
AdS-CFT
BCS-like Superconductor on the boundary!
QuickTime™ and aYUV420 codec decompressor
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Scale invariance versus BCS
J.-H. She
Gap equation:
1pp(kBT,,B) 0
Fermi-liquid:
Glue strength
Glue frequencySC gap
pp kBT, lnEF B
kBT,
kBTc Be1/, 2 3.5kBTc
Fermionic qu. critical:
Tc is large for moderate glue, gap to Tc ratio “universal” number!
pp kBT, 1
kBT
d2 pp
z
,1
i
d2 pp
z
kBTc B / 2B
c
2z
z
z
2z
, 2 A kBTc
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Cracking the Fermion signs: AdS-to-ARPES
Classical „Dirac waves‟
In progress ….
Schalm Sadri
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In conclusion …
Planckian dissipation: the beautiful hydrodynamic behaviors of quantum critical states of matter.
With help of string theory: this is in literal, but dual correspondence with the physics of black holes.
This sheds unexpected light on real life physics: the quark-gluon plasma and cold atoms.
AdS/CFT might well be the mathematics behind the enigma called High Tc superconductivity !
Further reading: Nature 430, 512 (2004), Science 315, 1372 (2007), Nature 448, 1000 (2007), Science 319, 1205 (2008); Hartnoll, Science 322, 1639 (2008)
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The dissipative world of apes …
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
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The fundamental constants
, c,G, e,
, c,G, e, kB
Boltzmann‟s constant is the conversion factor between time and probability !?
?
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56
In conclusion …
Planckian dissipation: the beautiful hydrodynamic behaviors of quantum critical states of matter.
With help of string theory: this is in literal, but dual correspondence with the physics of black holes.
This sheds unexpected light on real life physics: the quark-gluon plasma and cold atoms.
AdS/CFT might well be the mathematics behind the enigma called High Tc superconductivity !
Further reading: Nature 430, 512 (2004), Science 315, 1372 (2007), Nature 448, 1000 (2007), Science 319, 1205 (2008); Hartnoll, Science 322, 1639 (2008).
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empty
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Empty
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Empty
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Penrosian gravitational wave function collapse
Unitary/entangled/microscopic:
Dissipative/collapsed/macroscopic:
| Live > + | Dead >| Live > or | Dead >
id
dtLive
Live H Live
id
dtDead
Dead H Dead
d
dtLive
d
dtdead
Condition for unitarity: global time like Killing vector
Einstein gravity = „gauge theory of diffeomorphism‟: global time like killing vector does not exist in universes with different mass distributions!
Conflict becomes manifest at „collapse time‟:
coll.
G
L
GM 2
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Measuring quantum gravity
Penrosian collapse: ‘it takes a Schroedinger state made from e-coli’s one micrometer apart one second to collapse.’
Flux qubits closest approach with available technology!
MicroAmp currents, cross section 1 micron, height 0.1 mm:
Collapse time of order of seconds!
Van Wezel, Oosterkamp, JZ (cond-mat/0706.3976, Phil. Mag. B)
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The high Tc enigma
200oC
temperature
Quantum critical fermion matter
Electron correlations
conductor
Strangemetal
Super-
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Quantum Phase transitions
Quantum scale invariance emerges naturally at a zero temperature continuous phase transition driven by quantum fluctuations:
JZ, Science 319, 1205 (2008)
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Quantum critical QCD ??
Quantum phase transition ????
Quantum critical hydrodynamics
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The quark-gluon plasma
Relativistic Heavy Ion Collider
Quark-gluon ‘fireball’
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QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
boundary:
d-dim. Space-time
Hawking radiationgluons
strings
quarks
Black holes
High Tc superconductivity
Quark-gluon plasma
String theory: „AdS/CFT correspondence‟
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AdS/CFT dictionary:
E-field
D transverse E-field <=> D-1 electric field
D radial E-field <=> D-1 charge density
B-field
D radial B-field <=> D-1 magnetic field
D transverse B-field <=> D-1 current density
spatial metric perturb.
D transverse gradient <=> D-1 distortion
D radial gradient <=> D-1 stress tensor
temporal metric perturb.
D transverse gradient <=> D-1 temperature gradient
D radial gradient <=> D-1 heat flow
SUSY Einstein-Maxwell in AdS <==> SUSY Yang-Mills CFT
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Finite temperature fields: Euclidean Path Integrals
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Nernst effect: experiment
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AdS/CFT dictionary:
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
SUSY Einstein-Maxwell in AdS <==> SUSY Yang-Mills CFT
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Nernst effect: outcomes
Experiment Phuan Ong et al.
Qui ckTi me™ and a decompressor
are needed to see this pi cture.
AdS-CFT computation: Hartnoll, Kovtun, Mueller, Sachdev
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Superfluid-Mott insulator Quantum phase transition
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Optical conductivity near 2+1D superfluid-Mott
insulator QPT
g
SF
QC
MI
1()
79
Hartnoll, Herzog,Horowitz, arXiv:0803.3295
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Fermionic quantum phase transitions
JZ, Science 319, 1205 (2008)
„Mystery‟quantum critical metal (fermions !!)
JZ, Nature 430, 512 (2004)
High Tc superconductors
Heavy fermion metals
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Breaking SUSY CFT by chemical potentials …
g
QC
81
and supersymmetric!
Driving away from criticality by imposing finite fermion and boson charge density.Breaks scale invariance …
and supersymmetry!
Like fermion-boson model (?):
H bb f f (b f f f f b)
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Boson and Fermion pair optical conductivity
82
2 0
kBTc
8
Bose field:
Fermion pair field!
BCS like temperature dependence with
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Fermionic quantum criticality??
83
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84
Sachdev
She
Sadri
Schalm
Verlinde
McFadden
me
JZ: ‘A black hole full of answers’, Nature 448, 1001
(2007)
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The Viscosity-Entropy ratio Hartnoll-Son-Starinets (2002)
0L
1
2dtd
x e it Txy (t,
x ),Txy (0,0)
0L
q0L ImGxy,xy
R ,q
abs 16G
ImGR
abs 0 16G
s1
4
kB
CFT viscosity, Kubo formula:
AdS correspondent: graviton absorption cross section by black hole:
= area of horizon (GR theorems)
CFT entropy density: AdS correspondent = BH entropy = area of horizon
Universal viscosity-entropy ratio for CFT‟s with
gravitational dual:
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Conformal symmetry = quantum criticality
Supersymmetry <==> non-renormalization theorems „Planes of strongly interacting unstable fixed points‟AdS is the „geometrical dual‟ of conformal
invariance
Black holes, gauge background sources, etcetera, break conformal invariance
AdS/CFT is merely a „generating functional‟ of fanciful scaling
analysis ????
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87
Why do I care?
Roger Penrose Darius Sadri
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AdS/CFT for finite temperature hydrodynamics
Finite temperature fields: imaginary time compactification radius
R
kBT
Are there AdS spaces that are periodic in Euclidean time?
Yes, two (Hawking-Page, Witten):
(1)EAdS (low T confined phase, no entropy)
(2) Black hole in AdS (high T deconfined phase, entropy)
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‘Nernst regime’: fluctuating
superconducivityFinite temperature hydrodynamics of superconductor-insulator QCP??
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Ceperley path integrals and fermionic quantum
criticalityFermi-Dirac statistics encoded in ‘nodal surface constraints’ : probablistic theory.
Explicit example: Feynmannian Fermionic back flow, fractal nodal surface and quasiparticle effective mass divergence.
Krueger, JZ, arXiv:0804.2161
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The AdS/CFT challenge: fermionic quantum criticality
Heavy Fermion metals
JZ, Science 319, 1205 (2008)
High Tc Superconductors
JZ, Nature 430, 512 (2004)
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1
11
1
1
1
1
1
1
1
1
00
0 01
00
0
0 00
1
1
0
0
1
1
0
Hawking Temperature & Entropy
g = acceleration at horizon
A = area of horizon
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Strings and Black Holes
Universal physics of 0+1D Black hole <=> 1+1D CFT
● A classical = cosmic string is a Black string ● String theory has non-perturbative extended objects (like strings/vortices) called (mem)branes with p spatialdimensions
CLAIM:For certain (extremal) Black p-branes the universal (near horizon) brane physics is described by a p+1 dimensional CFT
[Maldacena]
The near-horizon of an extremal black brane is anAnti-de-Sitter space (homogeneous negative curvature)
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Viscosity/Entropy ratio
- Kubo formula = Zero frequency absorption of graviton by BH
- Zero frequency absorption of a graviton by BH = Area
- BH entropy = Area [Hawking]
Viscosity/Entropy ratio is order (1)!
[ In pQCD long calculation Visc/entropy ~ 1/g2ln(g)
[Arnold, Moore, Yaffe]]
[Policastro, Son, Starinets]
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kB
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Conformal symmetry = quantum criticality
AdS is the ‘geometrical dual’ of conformal invariance● Black holes, gauge sources break conformal invariance●
Supersymmetry <==> non-renormalization theorems●
● ‘Planes of unstable (strongly interacting) quantum critical
points’
AdS-CFT is ‘generating functional’ of fanciful scaling analysis ????
●
●
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Thermo-electric response coefficients
Kubo formula, e.g.
Hartnol, Kovtun
Muller, Sachdev
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(d+1)-dim Dyonic AdS Black Hole
metric + B-field fluctuations
linearized Einstein-Maxwell equation =>
two point functions of heat flow and electric currents.
thermo-electric coefficients follow from Kubo formula
dyonic AdS black holes exhibit Nernst effect!Hartnol, Kovtun
Muller, Sachdev
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(d+1)-dim AdS Black Hole = d-dim Thermal CFT
Black hole + metric fluctuations
linearized Einstein equation => quadratic Einstein action
= sufficient to compute two point functions of stress tensor.
Two point functions follow from free wave propagation
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AdS/CFT
Map of theories: map of 5d to 4d correlation functions
[Follows from open-closed string duality]
A Duality[Note: CFT sources are AdS fields] [Maldacena, Witten
Gubser, Klebanov,
Polyakov]
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String Theory for RHIC physics
● meson melting at high J● Mach cones due to heavy quarks
● Photon Production Rates● J/Psi screening● ...
Conceptually very simple calculations compared to (strong coupling) QCD
with answers very close to experiment (5-30%)
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Benefits from a natural point of view
AdS/CFT: ● CFTs are scale invariant
● CFTs: universal dynamics of continuous phase transitions
universal
...String theory for the real world?
universal
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Viscosity:
Kubo formula
Viscosity of the thermal conformal field theory
is computed from scattering of gravity waves off
an AdS black hole background
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Gauge/String Duality