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Gravity Theories and Their Avatars @CCTP, Crete July 13-19, 2012
Holographic Entanglement Entropy
from Cond-mat to Emergent Spacetime
Tadashi Takayanagi
Yukawa Institute for Theoretical Physics (YITP),
Kyoto University
Based on arXiv 1111.1023(JHEP 2012(125)1)
with N. Ogawa (RIKEN) and T. Ugajin (IPMU/YITP)
+ on going work with M.Nozaki (YITP) and S.Ryu (Ilinois,Urbana-S.)
Contents
①
②
③
④
⑤
⑥
Introduction
Entanglement Entropy and Fermi surfaces
Holographic Entanglement Entropy (HEE)
Fermi Surfaces and HEE
Emergent Metric via Quantum Entanglement
Conclusions
① Introduction– from cond-mat viewpoint
AdS/CFT is a very powerful method to understand strongly
coupled condensed matter systems.
Especially, the calculations become most tractable
in the strong coupling and large N limit of gauge theories.
In this limit, the AdS side is given by a classical gravity and
we can naturally expect universal behaviors such as the
no hair theorem in GR, η/s=1/4π, etc.
So, we concentrate on this limit for a while.
We would like to consider what is a universal properties for
metallic condensed matter systems via AdS/CFT.
The metals are usually described by the Landau’s Fermi
liquids. It is well-known that Fermi liquid states are stable
against perturbations by the Coulomb forces.
However, in strongly correlated
electron systems such as the
Strange Metal
strange metal phase of high Tc
superconductors or heavy fermion
Fermi Liquid
systems etc., we encounter
Pseudo Gap
so called non-Fermi liquids.
Mott Insulator
SC
Figs taken from Sachdev 0907.0008
So, one of the main purposes of this talk is to answer the question:
Can we obtain Landau’s Fermi liquids in the classical
gravity limit ?
Note: Several interesting setups of (non-)Fermi liquids have
already been found.
(i) Probe fermions in SUGRA b.g.
[Rey 07, Faulkner-Liu-McGreevy-Vegh 09, Cubrovic-Zaanen-Schalm 09,
DeWolfe-Gubser-Rosen 11, 12]
(ii) Electron stars (Lifshitz metric in the IR) [Hartnoll-Tavanfar 2010]
⇒ ∃Fermi surfaces, but, not in the leading order O(N2)
of the large N limit.
Systems with Fermi surfaces
⇔ Fermi liquids or non-Fermi liquids
So, we concentrate on systems with Fermi surfaces.
To make the presentation simpler, we will work for 2+1
dim. systems with Fermi surfaces. But our analysis can be
generalized to higher dimensions, straightforwardly.
ky
Fermi surface
Fermi sea
kx
How to characterize the Fermi surfaces ?
Metals ⇒ Conductivity ?
But it seems difficult to find universal results for conductivity in
the gravity dual. This is because it is related to the propagation
of U(1) gauge fields in AdS, whose behavior largely depends on
the precise Lagrangian of gauge fields e.g. f(φ)F2 .
So we want to find a quantity whose gravity dual is closely
related to the metric (i.e. gravity field).
⇒ We should look at a thermodynamical quantity !
One traditional candidate is the specific heat C.
For (Landau’s) fermi liquids, we always have the behavior
C  S  T V .
This linear specific heat can be understood if we note that we
can approximate the excitations of Fermi liquids by an infinite
copies of 2 dim. CFTs.
k //
E
Fermi surface
kF
E  k //
k 
 k y Fermi sea
 FL  
CFT
(
k
)
2
F

|k|
kF
Fermi surface
kx
In 2d CFT, we know
C  S  T  L.
In this way, we can estimate the specific heat of the Fermi liquids
C  S  T  L  ( Lk F )  k F  T  V .
However, the linear specific heat is not true for non-Fermi liquids.
This is because they have anomalous dynamical exponents z.
(~infinite copies of 2d Lifshitz theory: (t , x) ~ ( z t , x) )
C  S T
1/ z
V .
To characterize the existence of Fermi surfaces, we
need to look at a property which is common to both
the FL and non-FL.
⇒ The entanglement entropy is a suitable quantity.
After we concentrate on the systems with Fermi
surfaces, we can distinguish between FL and non-FL by
calculating the specific heat.
② Entanglement Entropy and Fermi surfaces
(2-1) Definition and Properties of Entanglement Entropy
Divide a quantum system
into two subsystems A and B:
Example: Spin Chain
H tot  H A  H B .
A
B
We define the reduced density matrix  A for A by
taking trace over the Hilbert space of B .
Now the entanglement entropy S A is defined by the
von-Neumann entropy
In QFTs, it is defined geometrically:
N : time slice
B A
A  B
(2-2) Area law
[Bombelli-Koul-Lee-Sorkin 86, Srednicki 93]
EE in QFTs includes UV divergences.
Area Law
In a (d+1 ) dim. QFT with a UV fixed point, the leading term of EE
is proportional to the area of the (d-1) dim. boundary A :
Area( A)
SA ~
 (subleading terms),
d 1
a
where
a
is a UV cutoff (i.e. lattice spacing).
Intuitively, this property is understood like:
Most strongly entangled
A
∂A
However, there are two known exceptions:
(a) 1+1 dim. CFT
B
A
B
[Holzhey-Larsen-Wilczek 94, Calabrese-Cardy 04]
(b) ∃Fermi surfaces (
)
A
[Wolf 05, Gioev-Klich 05]
(2-3) Fermi Surfaces and Entanglement Entropy
Why do Fermi Liquids violate the area law ?
This can be understood if we remember that the Fermi liquids
can be though of as infinite copies of 2d CFTs:
d 1
L
B
A
B
l
We will mainly assume this choice of subsystem A below.
Recently, there have been evidences that this logarithmic
behavior is true also for non-Fermi liquids (e.g.spin liquids).
[Swingle 09,10, Zhang-Grover-Vishwanath 11 etc.]
Intuitively, we can naturally expect this because the logarithmic
behavior does not change if we introduce the dynamical
z
exponent z in the 2d theory as log l  z log l.
Therefore we find the characterization:
∃Fermi surface ⇔ Logarithmic behavior of EE
To apply the AdS/CFT, we will embed the Fermi surface in a CFT.
In this case, the leading divergence still satisfies the area law.
But the subleading finite term has the logarithmic behavior:
Ld 1
d 1
S A    d 1    ( L  k F ) log(l  k F )  .
a
if we assume l  k F  1 .
d 1
L
B
A
B
l
So, we will concentrate on the gravity dual whose entanglement
entropy has this behavior in our arguments below.
③ Holographic Entanglement Entropy
Holographic Entanglement Entropy Formula
A
 A is the minimal area surface
(codim.=2) such that
A   A and A ~  A .
homologous
(We omit the time direction.)
CFTd 1
Area( A )
SA 
4G N
[Ryu-TT 06]
AdSd  2
B

za
(UV cut off)
 dt  i 1 dxi2  dz 2
2
2
2
ds AdS
 RAdS
z
d 1
z
2
.
• In spite of a heuristic argument [Fursaev, 06] , there has been no
complete proof. But, so many evidences and no counter examples.
[A Partial List of Evidences]




Area law follows straightforwardly [Ryu-TT 06]
Agreements with analytical 2d CFT results for AdS3 [Ryu-TT 06]
Holographic proof of strong subadditivity [Headrick-TT 07]
Consistency of 2d CFT results for disconnected subsystems
[Calabrese-Cardy-Tonni 09] with our holographic formula [Headrick 10]
 Agreement on the coefficient of log term in 4d CFT (~a+c)
[Ryu-TT 06, Solodukhin 08,10, Lohmayer-Neuberger-Schwimmer-Theisen 09,
Dowker 10, Casini-Huerta, 10, Myers-Sinha 10]
 A direct proof when A = round ball [Casini-Hueta-Myers 11]
 Holographic proof of Cadney-Linden-Winter inequality
[Hayden-Headrick-Maloney 11]
④ Fermi Surfaces and HEE
[Ogawa-Ugajin-TT 11]
(4-1) Setup of gravity dual
For simplicity, we consider a general gravity dual of 2+1 dim.
systems. The general metric can be written as follows (up to diff.)


2
R
ds 2  2  f ( z )dt 2  g ( z )dz 2  dx 2  dy 2 ,
z
where f(z) and g(z) are arbitrary functions.
We impose that it is asymptotically AdS4 i.e.
f ( z )  1 and
g ( z)  1
when z  0.
(4-2) Holographic EE
Now we would like to calculate the HEE for this gravity dual.
We choose the subsystem as the strip width l as before
x
L
B
A
y
B
l
x
AdS Bdy
the minimal surface condition reads
l
2

l
2
A
z*
z
In the end, we obtain
R2L
R2L
SA 
 kn
GN z F
2G N a
 l
 
 zF



n 1
n 1
 ...,
when the size of subsystem A is large l  z F .
In this case, the minimal surface extends to the IR region deeply.
⇒ The logarithmic behavior of EE is realized just when n=1.
 z
i.e.  Fermi Surface  g ( z ) 
 zF



2
( z  ) .
We identify z F as a characteristic scale of the Fermi energy.
Note: f(z) does not affect the HEE and is still arbitrary.
(4-3) Null Energy Condition
To have a sensible holographic dual, a necessary condition is
known as the null energy condition:
1

  
T N N   R  Rg   N N  0
2


for any null vector N  .


In the IR region, the null energy condition argues
g ( z)  z 2 ,
f ( z )  z 2 m 
m  1.
At finite temperature, we expect that the solution is given by
a black brane extension of our background:
2
2
2


dz
dx

dy
2
2
 2 ( m 1)
2
.
ds  R   z
h( z )dt  ~ 
2

z
h
(
z
)


The `non-extremal factors’ behave near the horizon
zH  z
h( z ) 
,
zH
z  zH
zH  z
~
h ( z) 
.
zH
From this, we can easily find the behavior of specific heat:
C  S T
2
m2
.
Combined with the null energy condition:
C T

m  1 , we obtain
2
with   .
3
Notice that this excludes the Landau’s Fermi liquids (α=1).
In summary, we find that classical gravity duals only allow
non-fermi liquids.
Comments:
(i) This result might not be so unnatural as the non-Fermi liquids
are expected in strongly correlated systems.
(ii) Even in the presence of perturbative higher derivative
corrections, the result does not seem to be changed.
(iii) Some miracle coincidences ?
AdS: No curvature singularity in the gravity dual
⇒ α=2/3 [11]
Shaghoulian
CMT: Spin fluctuations:
[Moriya, Hertz, Millis …. 70’-90’]
N Fermions + U(1) gauge:
⇒ α=2/3 (i.e. z=3)
[Lee 09, Metlitski, and S. Sachdev 10,
Mross-McFreevy-Liu-Senthil 10,
Lawler-Barci-Fernandez-Fradkin-Oxman 06]
Experiment: YbRh2(Si1-xGex)2
⇒ α=2/3
Examples of heavy fermions
[Pepin 11, talk at KITP]
(iv) We can embed this background in an effective gravity theory:
S EMS  161GN  dx d  2  g R  2  W ( ) F F       V ( )].
[Earlier works, Gubser-Rocha 09,
Charmousis-Goutéraux-Kim-Kiritsis-Meyer 10,…]
if W and V behave in the large φ limit as follows [Ogawa-Ugajin-TT 11]
( p 2  12 p  32) 
V ( )  2  
e
2
4 RAdS
W ( ) 
2
8A
e
2
2
z F p (8  p ) R
 f ( z)  z  p ,
3
2

( p 2)
g ( z)  z 2 ,
2

p2
,
,
(p  2 ).
(v) This metric can also be regarded as a generalization of Lifshitz
backgrounds so that it violates the hyperscaling.
[Huijse-Sachdev-Swingle 11, Dong-Harrison-Kachru-Torroba-Wang 12]
ds
2
( d  2)
r
 ( d  )
 r
2 ( z 1)
dt  dr  i 1 dxi2
2
2
d

.
 C  S  T ( d  ) / z .
d  1    d : S A ~ Lq , d-1  q  d  Violation of Area law
θ  d-1
:
0  θ  d-1 :
S A ~ (L) d-1 log L Fermi surface
S A ~ Lq , 0  q  d  1
`Landscape’ of (d+1) dim. Quantum Phases from HEE
Can we find examples
in realistic systems ?
Lower dimensional
structure
Power violation ?
(∃Hol. duals)
d-1
l
SA   
a
d
q
Not allowed
Area Law
(QFTs)
Logarithmic
Volume Law
(Fermi surface) (Non-local)
q
⑤ Emergent Metric from Quantum Entanglement
(5-1) Basic Outline
In principle, we can obtain a metric from a CFT as follows:
a CFT state ⇒ Information (~EE) = Minimal Areas ⇒ metric

SA
Area( A )
g 
One candidate of such frameworks is so called the entanglement
renormalization (MERA) [Vidal 05 (for a review see 0912.1651)] as
pointed out by [Swingle 09]. [cf. Emergent gravity: Raamsdonk 09, Lee 09]
(5-2) Tensor Network (TN)
[See e.g. Vidal 1106.1082 and references therein]
Recently, there have been remarkable progresses in numerical
algorithms for quantum lattice models, based on so called
tensor product states.
Basically, people try to find nice variational ansatzs for
the ground state wave functions for various spin systems.
⇒ An ansatz is good if it respects the quantum entanglement
of the true ground state.


M  ( )

Ex. Matrix Product State (MPS)
[DMRG: White 92,…,
Rommer-Ostlund 95,..]
1  2  3
n

1 2
n
 i  1,2,...,  ,
 i  or  .
Spin chain
 
Tr[ M ( ) M (

 

1
1 , 2 ,, n
2
)  M ( n )]  1 ,  2 ,,  n
n Spins
MPS and TTN are not good near quantum critical points (CFTs)
because their entanglement entropies are too small:
S A  2 log 
( log L ~ S
CFT
A
).
A
A

 1  2  3  n1  n
1  2  3  4  5  6  7
A
In general,
A
S A ~ N int  log  ,
N int  min[# Intersecti ons of  A ].
8
(5-3) AdS/CFT and (c)MERA
MERA (Multiscale Entanglement Renormalization Ansatz):
An efficient variational ansatz to find CFT ground states have been
developed recently. [Vidal 05 (for a review see 0912.1651)].
To respect its large entanglement in a CFT, we add (dis)entanglers.
Unitary transf.
between 2 spins
1  2  3  4  5  6  7
8
1  2  3  4  5  6  7
8
Calculations of EE in 1+1 dim. MERA
u  4
A= an interval (length L)
u  3
A
log L
u  2
u  1
A
L
S A  Min [# Bonds ]  log L
 agrees with 2d CFTs.
u0
A conjectued relation to AdS/CFT
[Swingle 09]
Min[# Bonds]
A
A
Min[Area]
u  ( u IR )
A
A
Equivalent ?
u


1
u0
AdSd  2
CFTd 1
2
2
e
dz  dt  dx
2
2
Metric  ds  2 (dt  dx ) 
,
2

z
where z    e u .
2u
2
2
Now, to make the connection to AdS/CFT clearer, we would like
to consider the MERA for quantum field theories.
Continuous MERA (cMERA)
[Haegeman-Osborne-Verschelde-Verstraete 11]
 (u )

u

 P  exp   i  ds[ K ( s )  L]  
u IR


True ground state
(highlyentangled)
Ω

,
IR state
(no entanglement)
 Real space renormaliz ation flow : length scale ~   e u .
K(s) : disentangler,
L: scale transformation
Conjecture
d  1 dim . cMERA  gravity on AdS d  2
z    e u .
(5-4) Emergent Metric from cMERA
[Nozaki-Ryu-TT, in preparation]
We conjecture that the metric in the extra direction is given by
using the idea of the quantum metric (up to a constant c):
g uu du  c 
2
1   (u ) | e
iLdu
Vol  
|  (u  du )
e
0
2
.
u
dk
d
Note: The quantum distance between two states is defined by
d ( 1 , 2 )  1   1 |  2
2
.
Comments
(1) The denominator represents the total volume of phase space
at energy scale u.
(2) The operation e^{iLdu} removes the coarse-graining
procedure so that we can measure the strength of
unitary transformations induced by disentanglers (bonds).
⇒ guu measures the density of bonds.
Consistent with the HEE:
0
S A ~  du g uu  e ( d 1)u
u IR
A
B
uUV  0
A
u   log z
u IR  
(5-5) Emergent Metric in a (d+1) dim. Free Scalar Theory
1
Hamiltonian: H   dk d [ (k ) (k )  (k 2  m 2 ) (k ) (k )].
2
: ak   0.
Ground state 
Moreover, we introduce the `IR state’  which has no real
space entanglement.
a x  M  ( x) 
a x   0,
i.e.
  0
x
 S A  0.
x
a x  M  ( x) 
i
M
i
M
 ( x),
 ( x).
For a free scalar theory, the ground state corresponds to




i
ˆ
K (u )   dk d  (u ) keu / M a k a k  (h.c.) ,
2
where (x) is a cut off function : (x)   (1- | x |).
1
e 2u
 ( s )   2u
, (for m  0,  (u)  1 / 2.)
2
2
2 e m /M
For the excited states,  (s ) becomes time-dependent.
One might be tempting to guess
Density of bonds
ds
2
Gravity
 g uu du 
2
e 2u

2
2
 dx  g tt dt 2
g ss  |  (u) | ?
2
g


(
u
)
.
Indeed, the previous proposal for guu lead to uu
Explicit metric
(i)
ds
2
Gravity
 g uu du 
2
e 2u
2
2
 dx  g tt dt 2
Massless scalar (E=k)
g uu
1

4
 the pure AdS
(ii) Lifshitz scalar (E=kν)
g uu 
2
4
 the Lifshitz geometry
(iii) Massive scalar
g uu 
4(e 2u
e 4u
.
2
2 2
m / )
2
2

 2
dz
1
m
2
 ds  2   2 2  2 dx  g tt dt 2
z
 
 z
Ｃａｐｐｅｄ ｏｆｆ ｉｎ ｔｈｅ ＩＲ
(5-6) Excited states after quantum quenches
( Ak ak  Bk ak )   0,
m(t)
m0
t
(| Ak |2  | Bk |2  1).
1/ 4
2
2 1/ 4
2


  ikt
1   k  m0 
k
   2

Ak  
e ,
2
2


2 
k

 k  m0  

1/ 4
2
2 1/ 4
2


  ikt
1   k  m0 
k
   2

Bk  
e .
2
2


2 
k

 k  m0  

Note: There is an phase factor ambiguity of (A,B).
⇒ Different choices of time slices ?
Time dependent metric from the Quantum Quench
Light cone
z g zz
looks like a propagation of
gravitational wave.
t
z
We can also confirm the linear growth of EE: SA∝t.
This is consistent with the known CFT (2d)
[Calabrese-Cardy 05]
and with the HEE results (any dim.). [Arrastia-Aparicio-Lopez 10 ,
Albash-Johnson 10, Balasubramanian-Bernamonti-de Boer-CoplandCraps- Keski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10, 11,….]
(5-7) Towards Holographic Dual of Flat Space
If we consider the (almost) flat metric
ds 2  e 2u du 2  e 2u dx 2  g uu  e 2u ,
the corresponding dispersion relation reads
1  k k Ek
 (u)   
2  Ek


 eu
 k eu
 Ek  e k .
This leads to the highly non-local Hamiltonian:
H   dx  ( x)e
d
 2
 ( x).
[cf. Li-TT 10]
⑤ Conclusions
• The entanglement entropy (EE) is a useful bridge between
gravity (string theory) and cond-mat physics.
•
Gravity
Entanglement
g 
S A  Area
Cond-mat.
systems 
Classical gravity duals + Null energy condition

with
⇒ a constraint on specific heat C  T
⇒ Non-fermi liquids !
2
 .
3
• Questions: String theory embeddings of the NFL b.g. ?
[See also solutions in Singh 10, Narayan 12]
• We studied the conjectured relation between AdS/CFT and
MERA. We especially employed the cMERA and proposed
a definition of metric in the extra dimension. This metric passes
several qualitative tests.
Many future problems: how to determine gtt ?
ambiguity of choices time slices ?
large N and strongly coupling limit ?
higher spin holography ?
:
: