Transcript Fermi liquid

AdS/CFT and the Fermions of
condensed matter
Jan Zaanen
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1
The Gross list: the 14 Big
Questions
1. The origin of the universe?
2. What is dark matter?
2004
11. What is space-time?
14. New states of matter: are there generic non-Fermi liquid states of
interacting condensed matter?
Solution of the Fermion sign problem??
?
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Photoemission
spectrum
‘Avatars of metallic phases’
Kachru
et al., arXiv:0909.2639
4
The ‘fermionic’ excitement
(a) Hydrodynamics: ‘Planckian dissipation’ (Hartnoll,
Sachdev, Harvard)
(b) The emergent (heavy) Fermi liquid (Schalm, Leiden)
(c) The ‘marginal-’, ‘critical’ non-Fermi-liquids (Liu, MIT)
(d) Holographic superconductivity (Hartnoll, Harvard)
5
Plan
1. Quantum critical electron systems and the great questions
of condensed matter physics
2. Hairy black holes versus zero temperature entropy in quantum
critical heavy fermion systems.
3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids.
6
Quantum critical matter
Quark gluon plasma
High Tc
superconductors
Heavy fermions
Metamagnetic
ruthenate
T
Tc
Quantum
critical
4
(P
2
nd
)
nt
poi
l
a
ic
crit
Tri
Li
ne
a
i ti c
cr
of
FM
2
o in
lp
ts
1s t
4
pq
0
3
1
2
2
3
1
P
7
H
The quantum in the kitchen:
Landau’s miracle
Kinetic energy
Electrons are waves
Fermi
energy
Pauli exclusion principle: every
state occupied by one electron
Fermi momenta
k=1/wavelength
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.
8
BCS theory: fermions turning into
bosons
Fermi-liquid + attractive interaction
Bardeen Cooper Schrieffer
Quasiparticles pair and Bose condense:
Ground state
D-wave SC: Dirac spectrum
 
BCS  k uk  v kck
ck  vac.
9
‘Shankar/Polchinski’ functional
renormalization group
UV: weakly interacting Fermi gas
Integrate momentum shells:
functions of running coupling
constants
interaction
Fermi sphere
All interactions (except marginal
Hartree) irrelevant => Scaling limit
might be perfectly ideal Fermi-gas
10
The end of weak coupling
interaction
Strong interactings:
Fermi gas as UV starting point
does not make sense!
=> ‘emergent’ Fermi liquid fixed
point remarkably resilient (e.g. 3He)
Fermi sphere
=> Non Fermi-liquid/non ‘HartreeFock’ (BCS etc) states of fermion
matter?
11
Fermion sign problem
Imaginary time path-integral formulation
Boltzmannons or Bosons:
Fermions:
 integrand non-negative
 negative Boltzmann weights
 probability of equivalent classical
system: (crosslinked) ringpolymers
 non probablistic: NP-hard
problem (Troyer, Wiese)!!!
Fractal Cauliflower (romanesco)
Quantum critical cauliflower
Quantum critical cauliflower
Quantum critical cauliflower
Quantum critical cauliflower
Quantum criticality or ‘conformal
fields’
18
Quantum Phase transitions
Quantum scale invariance emerges naturally at a zero temperature
continuous phase transition driven by quantum fluctuations:
JZ, Science 319, 1205 (2008)
19
Fermionic renormalization
group
The Magic of AdS/CFT!
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boundary:
d-dim space-time
strings
Black holes
Wilson-Fisher RG:
based on Boltzmannian
statistical physics
gluons
quarks
Hawking radiation
20
The ‘fermionic’ excitement
(a) Hydrodynamics: ‘Planckian dissipation’
(b) The emergent (heavy) Fermi liquid
(c) The ‘marginal-’, ‘critical’ non-Fermi-liquids
(d) Holographic superconductivity
21
Quantum critical matter
Quark gluon plasma
High Tc
superconductors
Heavy fermions
Metamagnetic
ruthenate
T
Tc
Quantum
critical
4
(P
2
nd
)
nt
poi
l
a
ic
crit
Tri
Li
ne
a
i ti c
cr
of
FM
2
o in
lp
ts
1s t
4
pq
0
3
1
2
2
3
1
P
22
H
Quantum criticality and Planckian

dissipation
Quark-gluon plasma: ‘minimal viscosity’ => absorption cross
section of gravitons by the black hole (AdS/CFT)
s 4 k B

 1
 T
Quantum criticality => ‘Quantum limit of dissipation’:
relaxation (= entropy production) time

  
kB T
Linear resistivity in cuprates governed by ‘Planckian’
relaxation
Van der Marel et al., Nature 425, 271 (2003); JZ, Nature 430,
(2004), Cooper et al., Science 323, 603 (2009)
512
23
Dissipation = absorption of
classical waves by Black hole!
Hartnoll-Son-Starinets (2002):

Viscosity: absorption cross section of
 abs0
gravitons by black hole

16G
= area of horizon (GR theorems)
Entropy densitys: Bekenstein-Hawking
BH entropy = area of horizon
Universal viscosity-entropy ratio for CFT’s
with gravitational dual limited in large N by:
s 4 k B

 1
24
The quark-gluon plasma
Relativistic Heavy Ion Collider
Quark-gluon ‘fireball’
25
The tiny viscosity of the QuarkGluon plasma
4kB
s
QG plasma:
within
 20% of
the AdS/CFT
viscosity!
Quantum critical hydrodynamics:
Planckian relaxation time
Relaxation time  : time it takes to convert
work in entropy.
    p
 p
s
T
Viscosity:
Entropy
 density:
“Planckianviscosity”


s
 T
kB T
kB
??
1


   

kB T
Planckian relaxation time = the shortest possible
 conditions that can
relaxation time under equilibrium
only be reached when the quantum dynamics is scale
invariant !!
27
Twenty three years ago …
Mueller
Bednorz
Ceramic CuO’s,
likeYBa2Cu3O7
Superconductivity
jumps to ‘high’
temperatures
28
Graveyard of Theories
Mott
De Gennes
Mueller
Laughlin
Bednorz
Anderson
Wilczek
Ginzburg
Abrikosov
Schrieffer
Leggett
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Lee
Yang
29
Phase diagram high Tc
superconductors
‘Stripy stuff’, spontaneous
currents, phase fluctuations ..
Mystery
quantum critical
metal
The return of
normalcy

BCS  k uk  v kck
ck  vac.
 
JZ, Science 315,
1372 (2007)
30
Divine resistivity
31
Critical Cuprates are Planckian
Dissipators
van der Marel, JZ, … Nature 2003:
Optical conductivity QC cuprates
Frequency less than temperature:
2
1  pr r
1(,T) 
, r  A
2 2
4 1   r
kB T

[
]  const.(1 A [
2
kB T1

kB T
]2 )
A= 0.7: the normal state of optimallly doped cuprates is a
Planckian dissipator!
32
Divine resistivity
?!
33

Planckian dissipation
Quark gluon plasma
4 k B

 T
 1
s
Viscosity bound
Planckian relaxation time:
High Tc
superconductors
 
kB T
Linear resistivity normal state (JZ, Nature 430,512) :
T  
Quantum
critical
1 1
  
2
p
   0.7
kB T
Tc is set by competition between superconductor
and critical normal state (e.g. arXiv:0905.1225)

34
Why the Wilsonian renormalization
group fails …
Superconductorinsulator QCP
PRL 95, 107002 (2005)
Chamon
Phillips
(a) Charge conservation (‘hydrodynamics’) imposes engineering
scaling dimensions on current
(b) Scale invariance: assume one diverging length scale.
d 2
z
e kB T 
 ,T    
 c 
2
1
 DC  ??
T
d 2
z
  
kB T 
e
    DC T   0 
 c 
kB T 
2
d =2 or 3 implies that z < 0 !?
35
The ‘fermionic’ excitement
(a) Hydrodynamics: ‘Planckian dissipation’
(b) The emergent (heavy) Fermi liquid
(c) The ‘marginal-’, ‘critical’ non-Fermi-liquids
(d) Holographic superconductivity
36
Watching electrons:
photoemission
Kinetic energy
Fermi
energy
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Electron spectral function: probability to
create or annihilate an electron at a
given momentum and energy.
Fermi momenta
k=1/wavelength
Fermi
energy
energy
Fermi surface of copper
k=1/wavelength
37
Fermi-liquid
phenomenology
Bare single fermion propagator ‘enumerates the fixed point’:
G  , k  
Spectral function:
1
Z

  0  k 2 2m    i   EF   vR k  k F   
ImG(,k)  A,k  
,k 
2
    k  k F  2m  ,k   ,k 
2
2
The Fermi liquid ‘lawyer list’:
- At T= 0 thespectral weight is zero at the Fermi-energy except for the
quasiparticle peak at the Fermi surface: AE F ,k   Z  k  kF 
- Analytical structure of the self-energy: ,k     E F  
2
 , k   EF , k F  
- Temperature dependence:


  EF    k  k F   
   EF
k k  k F

E F ,kF ,T T 2 

38
ARPES: Observing Fermi liquids
‘MDC’ at EF in conventional
2D metal (NbSe2)
Fermi-liquids: sharp Quasiparticle ‘poles’
39
Cuprates: “Marginal” or “Critical”
Fermi liquids
Fermi ‘arcs’ (underdoped)
closing to Fermi-surfaces
(optimally-, overdoped).
EDC lineshape: ‘branch cut’ (conformal),
width propotional to energy
40
Varma’s Marginal Fermi liquid
phenomenology.
Fermi-gas interacting by second order perturbation theory with ‘singular heat bath’:

ImP(q, )  N(0) , for |  | T
T
 N(0)sign  , for |  | T
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Directly observed in e.g. Raman ??
Single electron response 
(photoemission):
G(k, ) 
1
  v F k  kF   (k, )
2
 g  


(k, )     ln max |  |,T / c  i max |  |,T 

2
 c  

1
Single particle life time  max |  |,T is coincident (?!) with the

transport life time => linear resistivity.

41
Senthil’s critical Fermi-surface
arXiv:0803.4092
Single particle spectral function:
 c1 
 k//  
A K,,T,g   k//  / zk//  F  zk//  , ,k  g  gc

T

k




c0
Time like: truly conformal
(branch cuts)
Space like: still a ‘critical’ Fermi
surface (non conformal)
Fermi-surface: higher order
singularity in n(k)
42
Critical fermions at zero density:
branchcut propagators
 ,r     ,r   (0,0)
Two point Euclidean correlators:
Analytically continue to Minkowski time => susceptibilities  t,r   i ,r 

At criticality, conformal invariance:
Lorentz invariance:
 ,k  


  
1
1

 2  c 2 k 2





1


n
  ( ) 
1
i 

Scaling dimension set
by mass in AdS Dirac
equation.

43
Quantum critical matter
Quark gluon plasma
High Tc
superconductors
Heavy fermions
Metamagnetic
ruthenate
T
Tc
Quantum
critical
4
(P
2
nd
)
nt
poi
l
a
ic
crit
Tri
Li
ne
a
i ti c
cr
of
FM
2
o in
lp
ts
1s t
4
pq
0
3
1
2
2
3
1
P
44
H
Quantum critical
 transport in
heavy fermion systems

Blue = Fermi liquid:
 T2
Yellow= quantum
critical regime:
 T
Antiferromagnetic
order
Custers et al., Nature
(2003)
Fermions and Hertz-MillisMoriya-Lonzarich
Ideologically like marginal FL:
Bosonic (magnetic, etc.) order
parameter drives the phase transition
Supercon
ductivity
Fermi gas
Electrons: fermion gas = heat bath
damping bosonic critical fluctuations
Bosonic critical fluctuations ‘back react’
as pairing glue on the electrons
46
Fermionic quantum phase transitions
in the heavy fermion metals
JZ, Science 319, 1205
(2008)
QP effective mass
1
m 
EF
*
E F  0  m*  
‘bad
actors’
Paschen et al., Nature (2004)
Coleman
Rutgers
Critical Fermi surfaces in heavy
fermion systems
Blue = Fermi liquid
Yellow= quantum
critical regime
Antiferromagnetic
order
FL Fermi surface
Coexisting critical
Fermi surfaces ?
FL Fermi surface
Plan
1. Quantum critical electron systems and the great questions
of condensed matter physics
2. Hairy black holes versus zero temperature entropy in
quantum critical heavy fermion systems.
3. Fermion signs versus the AdS/CFT emergent (non) Fermi-liquids.
49
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d eco mpres sor
are nee ded to s ee this picture .
Hartnoll Herzog
Spielberg
Thorne
Nature Nov 5 2009
Horowitz
Fisk
Grigeria MacKenzie ThomsonRonning
St Andrews
Los Alamos
50
The zero temperature extensive
entropy ‘disaster’
The ‘extremal’ charged black
hole in AdS has zero Hawking
temperature but a finite
horizon area.
AdS-CFT
The ‘seriously entangled’
quantum critical matter at
zero temperature should have
an extensive ground state
entropy (?*##!!)
51
The holographic superconductor
Hartnoll, Herzog, Horowitz, arXiv:0803.3295
Condensate (superconductor,
… ) on the boundary!
AdS-CFT
(Scalar) matter ‘atmosphere’
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‘Super radiance’: in the
presence of matter the
extremal BH is unstable =>
zero T entropy always
avoided by low T order!!!
52
Hair as vacuum polarization
53
Fermionic quantum phase transitions
in the heavy fermion metals
JZ, Science 319, 1205
(2008)
QP effective mass
1
m 
EF
*
E F  0  m*  
‘bad
actors’
Paschen et al., Nature (2004)
Coleman
Rutgers
The metamagnetic quantum
critical end point
T
Tc
4
(P
2n d
)
int
o
p
l
tica
i
r
c
Tri
Ends at critical end point
a
i ti c
cr
of
FM
Mackenzie
Metamagnetism: first order jump from
low to high magnetization in field =
transition between water and steam
Li
ne
2
Grigera
o in
lp
ts
1s t
4
pq
0
3
1
2
2
3
1
H
‘St. Andrews’ ruthenate (Sr3Ru2O7):
critical end point can be tuned to
zero temperature
P
55
The ‘entropic singularity’
(Rost et al., Science 325, 1360, 2009)
lim T 0
S
1

1
T
H  H c 
Zero T entropy ??
QPT
56
The naked singularity always gets
covered up …
Exotic quantum nematic ‘hiding’
the critical point.
1.4
2
1.0
T[K]
‘Naked’ quantum critical point
dR/dT
1.2
Perspective Z. Fisk
0.8
0
0.6
0.4
QuickTime™ and a
d eco mpres sor
are nee ded to s ee this picture .
Science 325, 1348 (2009)
-2
0.2
0.0
7
8
h[T]
9
‘The unusual new phase can be thought of as
the material’s solution to the problem of
lowering its entropy in accord with the third
law of thermodynamics’
57
Experimentalists: back to the
entropic drawing board ..
Ruthenates:
St. Andrews
T
Tc
4
(P
2n d
)
Lanthanides, actinides:
Los Alamos
int
l po
ica
crit
i
r
T
Li
ne
a
i ti c
cr
of
FM
2
o in
lp
ts
1s t
4
pq
0
H
3
1
2
2
3
1
P
Nailing down T=0 entropy hidden
by last minute order: high
precision entropy balance needed.
Sorder 
Tc

0
C
dT
T
Grigeria MacKenzie
Thomson Ronning
58
Plan
1. Quantum critical electron systems and the great questions
of condensed matter physics
2. Hairy black holes versus zero temperature entropy in quantum
critical heavy fermion systems.
3. Fermion signs versus the AdS/CFT emergent (non) Fermiliquids.
59
“AdS-to-ARPES”: Fermi-liquid
emerging from a quantum critical state.
Cubrovic
Schalm
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decompressor
are needed to see this picture.
60
Fermionic quantum phase transitions
in the heavy fermion metals
JZ, Science 319, 1205
(2008)
QP effective mass
1
m 
EF
*
E F  0  m*  
‘bad
actors’
Paschen et al., Nature (2004)
Coleman
Rutgers
The fermionic criticality conundrum
Kinetic energy
Fermi
energy
Pauli exclusion principle generates
the Fermi-energy, Fermi surface.
How to reconcile the quantum
statistical scales with scale invariance?
Fermi momenta
k=1/wavelength
Fermi surface of copper
How can a (heavy) Fermi-liquid
emerge from a ‘microscopic’
quantum critical state?
AdS/CFT gives an answer!
62
Breaking fermionic criticality
with a chemical potential
Classical ‘Dirac waves’
E
Fermi-surface??



Electrical monopole
k
63
Fermion spectral function from RN
Black hole (e.g. McGreevy, arXiv:0909.0518)
Bulk Theory: Einstein-Maxwell + charged fermions
S
G
R ,k  :



1
gR  6  F 2  eAM  A DM  igAM   m Sbnd


4
Solve Dirac equations of motion, infalling boundary condition at ReissnerNordstrom black hole horizon (horizon at z=1, boundary at z=0)

Reissner-Nordstrom
BH:
4T  3 q2 , 0  2q
2
2
2
2
2
ds 

z2
 f zdt  dx1  dx 2 
1 dz
f z z 2
A0  2q z 1
f z  1 zz 2  z  1 q 2 z 3 
1. Universal: only depends on gq and m for any theory
2. Caveat: spectral function is a probe, possible dynamical instability ignored.

64
“AdS-to-ARPES”:
emergent Fermi-liquid.
Lorentz + Scale invariance (branchcut):
 
1


Schalm Cubrovic
1

  c k
2
2 2


2

QP peak!
Emergent heavy Fermi liquid
Scale invariance (branchcut)
65
Fermi-liquid
phenomenology
Bare single fermion propagator ‘enumerates the fixed point’:
G  , k  
Spectral function:
1
Z

  0  k 2 2m    i   EF   vR k  k F   
ImG(,k)  A,k  
,k 
2
    k  k F  2m  ,k   ,k 
2
2
The Fermi liquid ‘lawyer list’:
- At T= 0 thespectral weight is zero at the Fermi-energy except for the
quasiparticle peak at the Fermi surface: AE F ,k   Z  k  kF 
- Analytical structure of the self-energy: ,k     E F  
2
 , k   EF , k F  
- Temperature dependence:


  EF    k  k F   
   EF
k k  k F

E F ,kF ,T T 2 

66
“AdS-to-ARPES”: the
quasiparticles.
AE F ,k   Z  k  kF  ??

Schalm Cubrovic
Dip in the
spectrum at
E F ,kF 

67
“AdS-to-ARPES”:
quasiparticle life times.
Schalm Cubrovic
Temperature dependence:
Energy dependence self-energy:
No constant term in  "(EF ,k)

"E F ,kF ,T  T 2
T  T 
R
F

68
The AdS/CFT heavy
Fermi liquid
Landau Fermi-liquid
Quasiparticle mass grows indefinitely when
allowed by broken Lorentz invariance
69
“AdS-to-ARPES”:
summary.
Schalm Cubrovic
QP mass increases indefinitely
in the emergent Fermi-liquid
AdS
density
Fermionic quantum phase
transition at finite density
where the Fermi surface
disappears
Scaling dimension Fermion fields
70
Spectral function:
overview
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Non-Fermi-liquid
  0.1

Landau Fermi-liquid
 1
0

71
The emergent
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2
horizon AdS
Hong Liu (MIT)
Horizon geometry of the extremal
black hole: ‘emergent’ AdS2 =>
IR of boundary theory controlled
by emergent CFT1(Hong Liu)

Gravitational ‘mechanism’ for marginal
(critical) Fermi-liquids:
G1    vF k  kF   k,
Spatial: Fermi-surface implied by matching
UV-IR
"
2 kF
Temporal: damping controlled by AdS2,
relevancy implies non Fermi-liquid

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72
Gravitationally coding the fermion
propagators (Faulkner et al. arXiv:0907.2694)
T=0 extremal black hole, near horizon geometry ‘emergent scale invariant’:
AdS2  R2  gk   ck
2 k
Matching with the UV infalling Dirac waves:
GR ,k  F0 k  F1k  F2 (k)gk 
Special momentum shell: | k | kF

h1
i
2
GR (,k) 
; ,k   hgkF    h2e kF  kF
 k  kF   /v F  ,k 
Space-like: IR-UV matching ‘organizes’ Fermi-surface.

Time-like: IR scale invariance picked up via AdS2 self energy
Miracle, this is like critical/marginal Fermi-liquids!!
73
The emergent
horizon AdS2
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arXiv:0907.2694

CFT Fermion propagator:
k  kF   /v F  ,k 
kF    h2e


GR (,k) 
;


,k

hg
 kF
i kF
2
h1
k
F
m 2 1 kF2 1



2
6 2 3
Fermi-liquids:
AdS2

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
 k F  1/2 :
2 kF
  k  kF , " k  k F 
‘Critical Fermi liquids with Fermi surface:
 k F  1/2 :
  (k  kF )
1
2 kF
, " k  kF 
74
1
2 kF
Two families of emergent fermion
infrareds
Fermi-liquid
Emergent “CFT1” critical non-Fermi-liquids
see Liu et al., arXiv:0907.2694
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",kF    2
(reproduced in Leiden)
“Marginal” Fermi-liquid
" ,kF     ,
 1
“Pseudogap” phase
Z   0,k   0, k
75
Thermodynamics: where are the
fermions?
Hartnoll et al.: arXiv:0908.2657,0912.0008
Large N limit: thermodynamics entirely determined by
AdS geometry.
Fermi surface dependent thermodynamics, e.g. Haas van
Alphen oscillations?
Leading 1/N corrections: “Fermionic one-loop
dark energy”
Quantum corrections: one loop using Dirac quasinormal modes:
‘generalized Lifshitz-Kosevich formula’ for HvA oscillations.
 2osc. ATckF4
ck F2
 osc.  

cos
2
3
B
eB
eB

e
cTk F2

eb
T 2 1
Fn  
 
 
n 0
76
Collective transport: fermion
currents
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Hong Liu (MIT)
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decompressor
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Tedious one loop calculation, ‘accidental’ cancellations:
FS  "1 fermion  T 2
‘Strange coincidence’ of one electron and transport lifetime of marginal fermi
liquid finds gravitational explanation!

77
Soaked in Entropy ….
Entropic catastrophe!
S  A  C Td 
F AT
78
The unstable RN Black hole:
Fermion VEV’s!
Backreaction of matter: need VEV’s/classical finite amplitude fields in AdS.
Cannot be formed from single fermion fields, but bilinears:
J     ??
Witten: multi trace operators in AdS/CFT (hep-th/0112258)
Fermion density J0 could be finite !?
Back reaction on scalar potential A0, keep RN-AdS
 geometry fixed:
   , ,J  J  J ,I0   
i


B


T

 z   i   0
T
  gA z, k

f z
i
0

f z


z
z
 2 B J 0  2T0 0 I 0  0
 2 B0 I 0  T0 0 J 0  J 0   0
 zz A0 
gJ 0
0
3
z f z 
79
The dictionary and Fermion
VEV’s
How do bilinears like
J    
The fields near the boundary (z -> 0):
fit in the dictionary?
   z   z


Proposal: 
The bilinears should go simply like the fields squared at the tree
level
J z

0
2 
, J z

0
2 
Following Witten’s proposal for multi trace operators in AdS/CFT (hepth/0112258)
80
Fermionic density back hole hair!
Density switches on!
First order gas-liquid
transition at ‘high’
temperature: strongly
interacting matter!
81
Fermionic hair and the Fermi liquid
stability.
Strongly renormalized EF
Single Fermion spectral function:
pathological temperature dependent
spectral weight transfer removed.
82
The fermion probe limit as a lucky
accident
E z
Ez
z 
The change in BH electrical
field is numerically very small
But the field is diverging near the
horizon
A0      log 1  z 1  z 
The near-horizon geometry will
backreact as well!
83
Luttinger volume versus bulk
density
Luttinger theorem: the volume in k
space enclosed by the Fermi surface
of a Fermi liquid does not
renormalize.
Given kF (from the spectral functions)
the density is:
q
 2
k F
This is within numerical accuracy
equal to the CFT density following
from the bulk fermion VEV’s!
84
Serious fermion hair: the Lifshitz
horizon
Divergence of E field at horizon: geometry has to backreact !
Solve equations of motion for Maxwell, Dirac and Einstein
z f  0, z   0, zz A0  0, z   0
Metric:
2
2
2




dt
exp


z
dx

dy
ds2 
 dz 2 f z  
f z 
z2

As for holographic superconductor this turns into a Lifshitz geometry (->
Horova):
Lifshitz exponent
ds 
2
dt 2 exp  z
z 2
dx 2  dy 2
 dz z 
z2
2 2
85
Full backreaction: the stability of
the Fermi-liquid!
Non Fermi-liquid Fermi surfaces disappear from the spectral functions!
Perfect Fermi-liquid
86
Holographic superconductivity:
stabilizing the fermions.
Fermion spectrum for scalar-hair back hole (Faulkner et al., 911.340;
Chen et al., 0911.282):
Temperature dependence as expected for
‘quantum-critical’ superconductivity (She,
JZ, 0905.1225)
‘BCS’ Gap in fermion
spectrum !!
Excessive temperature dependence
‘pacified’ !
87
‘Pseudogap’ fermions in high Tc
superconductors
102 K
Tc = 82 K
Gap stays open above Tc
10 K
But sharp quasiparticles
disappear in incoherent
‘spectral smears’ in the metal
Shen group, Nature 450, 81 (2007)
88
The moral of this story…
The AdS/CFT correspondence: processing the fermion signs in
fermionic quantum critical states
The mystery of Fermi-liquid stability: in the ‘hairy’ gravitational
dual it is just like a bosonic condensate!!
Critical Fermi-liquid phenomenology: AdS2 horizons needed,
do they occur at ‘isolated’ quantum critical points associated
with bosonic order?
Holographic superconductors feeds on the AdS2 T=0 entropy:
employment program for condensed matter experimentalists!
89
Further reading
AdS/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; K. Schalm et al.
Science, and to appear any moment; T. Faulkner et al., arXiv:
0911.3402
Condensed matter:
High Tc: J. Zaanen et al., Nature 430, 512, Nature Physics 2, 138; C.M.
Varma et al., Phys. Rep. 361, 267417
Heavy Fermions: J. Zaanen, Science 319, 1205; von Loehneisen et al, Rev.
Mod. Phys. 79, 1015
90
Thermodynamics from classical
action
Total action:
S  Sbulk  Sbnd
Bulk action: equation of motions
Sbulk  
Boundary action
1 2


 g  p   m  F 
4 

Sbnd    h    F A 
For fermions bulk action vanishes in the classical limit Sbulk  0
But boundary action is finite => Free energy is determined by I0 and J0
Sbnd  I 0   J 0  F0
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