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.)

Gravity Theories and Their Avatars @CCTP, Crete　 July 13-19, 2012

Contents

① 　 Introduction ② 　 Entanglement Entropy and Fermi surfaces③ Holographic Entanglement Entropy (HEE)④ Fermi Surfaces and HEE ⑤ 　 Emergent Metric via Quantum Entanglement⑥ 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 phase of high Tc superconductors or heavy fermion systems etc., we encounter so called non-Fermi liquids.

Pseudo Gap

Strange Metal

Fermi Liquid

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.

(ii) Electron stars (Lifshitz metric in the IR) ⇒ 　∃ Fermi surfaces, but, not in the leading order O(N2) of the large N limit.

[Hartnoll-Tavanfar 2010]

[Rey 07, Faulkner-Liu-McGreevy-Vegh 09, Cubrovic-Zaanen-Schalm 09, DeWolfe-Gubser-Rosen 11, 12]

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.

xk

ykFermi surface

Fermi sea

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

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.

.VTSC

|| k

EFermi surface

Fk

FFk

kE

k2

//

)(CFT FL

//k

k

Fermi surfacexk

ykFermi sea

In 2d CFT, we know

In this way, we can estimate the specific heat of the Fermi liquids

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: )

.LTSC

.)( VTkLkLTSC FF

./1 VTSC z

),(~),( xtxt z

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.

Divide a quantum system into two subsystems A and B:

We define the reduced density matrix for A by

taking trace over the Hilbert space of B .

A

, Tr totBA

. BAtot HHH

A BExample: Spin Chain

② Entanglement Entropy and Fermi surfaces 　(2-1) Definition and Properties of Entanglement Entropy

Now the entanglement entropy is defined by the von-Neumann entropy

In QFTs, it is defined geometrically:

. log Tr AAAAS

AS

BA B A

slice time:N

(2-2) Area law

EE in QFTs includes UV divergences.

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 :

where is a UV cutoff (i.e. lattice spacing).

Intuitively, this property is understood like:

Most strongly entangled

terms),subleading(A)Area(~ 1 dA a

S

A

a

[Bombelli-Koul-Lee-Sorkin 86, Srednicki 93]

A∂A

Area Law

However, there are two known exceptions:

(a) 1+1 dim. CFT

(b) Fermi surfaces ( )∃ 1~ akF

.log3 a

lcSA

...log~1

al

alS

d

AlA

B A B

l

[Holzhey-Larsen-Wilczek 94, Calabrese-Cardy 04]

[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:

We will mainly assume this choice of subsystem A below.

CFT 2d ofEach CFTs 2d of #

1

.log~al

aLS

d

A

1dL B A B

l

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 exponent z in the 2d theory as .

Therefore we find the characterization:

∃Fermi surface　⇔　 Logarithmic behavior of EE

lzl z loglog

.)log()( 11

1

Fd

Fd

d

A klkLaLS

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:

if we assume .

So, we will concentrate on the gravity dual whose entanglement entropy has this behavior in our arguments below.

1dL B A B

l

1 Fkl

Holographic Entanglement Entropy Formula [Ryu-TT 06]

is the minimal area surface (codim.=2) such that

homologous

. ~ and AA AA

N

AA 4G

)Area(S

2dAdS z

A

)direction. timeomit the (We

③ Holographic Entanglement Entropy

BA

1dCFT

. 2

1

1222

22

zdzdxdt

Rdsd

i iAdSAdS

off)cut (UV

az

• 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.)

where f(z) and g(z) are arbitrary functions.We impose that it is asymptotically AdS4 i.e.

,)()( 22222

22 dydxdzzgdtzf

zRds

.0n whe1)( and 1)( zzgzf

(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

the minimal surface condition reads

L B A B

l

yx

LylxlyxA 0 ,

22|),(

.)(')(2Area* 2

22

z

azxzg

zdzLR

zAd

S Bd

y Ax

2l

2l

*z.

/1)()(' 4

*42

*

2

zzzg

zzzx

In the end, we obtain

when the size of subsystem A is large . In this case, the minimal surface extends to the IR region deeply.

⇒　 The logarithmic behavior of EE is realized just when n=1.

We identify as a characteristic scale of the Fermi energy.Note: f(z) does not affect the HEE and is still arbitrary.

...,2

11

22

nn

FFNn

NA z

lzGLRk

aGLRS

Fzl

Fz

. )( )( Surface Fermi ..2

z

zzzgeiF

(4-3) Null Energy Condition

To have a sensible holographic dual, a necessary condition is known as the null energy condition:

In the IR region, the null energy condition argues

. vector nullany for

021

N

NNRgRNNT

.1 )( ,)( 22 mzzfzzg m

At finite temperature, we expect that the solution is given by a black brane extension of our background:

The `non-extremal factors’ behave near the horizon

From this, we can easily find the behavior of specific heat:

.)(

~)( 2

2222)1(222

zdydx

zhdzdtzhzRds m

.)(~

,)(H

H

H

H

zzz

zhz

zzzh

Hzz

. 22 mTSC

Combined with the null energy condition: , we obtain

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.

.32 with TC

1m

(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:

[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]

)].()(2216

1

VFFWRgdxS d

GEMS N

). (pzzgzzf

eRppz

AW

eR

ppV

p

p

F

p

AdS

2 ,)( ,)(

, )8(

8)(

,4

)3212(2)(

2

)2(23

22

2

22

2

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]

.

. /)(

1222)1(2)(

)2(2

zd

d

i izd

d

TSC

dxdrdtrrds

10 ,L ~ : 10

surface Fermi log~ : 1

law Area ofViolation 1 ,L ~ : 11

dqS d-θ

L (L)S d-θ

d q d-Sdd

qA

d-A

qA

`Landscape’ of (d+1) dim. Quantum Phases from HEE

q

A alS

qd-1 d

Area Law (QFTs)

Logarithmic(Fermi surface)

Volume Law (Non-local)

Lower dimensional structure

Not allowed

Power violation ? ( Hol. duals)∃

Can we find examples in realistic systems ?

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

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].

AS )Area( A g

[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.

Ex. Matrix Product State (MPS) [DMRG: White 92,…, Rommer-Ostlund 95,..]

)(M

1 2 n

1 23 n

Spinsnnn

n

MMM 21

,,,21 ,,, ])()()(Tr[

21

. or

,1,2,...,

i

i

Spin chain

MPS and TTN are not good near quantum critical points (CFTs) because their entanglement entropies are too small:

In general,

).~log( log2 CFTAA SLS

1 2 n1 2 3 4 6 7 853 1n

A A

]. of onsIntersectimin[#,log~

int

int

A

A

NNS

A A

(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.

1 2 3 4 6 7 85 1 2 3 4 6 7 85

Unitary transf.between 2 spins

Calculations of EE in 1+1 dim. MERA

A= an interval (length L)

A

A

CFTs. 2d with agrees log]Bonds[#Min

LS A

L

Llog

0u

1u2u

3u

4u

A conjectued relation to AdS/CFT 　 [Swingle 09]

0u 1u

A A

A

Min[Area]

A

.

,)(Metric 2

22222

2

22

u

u

ezwherez

xddtdzxddteds

2dAdS1dCFT

A

Bonds]Min[#

? Equivalent

)( IRuu

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]

. AdSon gravity cMERA .dim 1

. ~ scalelength :flowation renormaliz space Real

,Ω])([exp)(

2d

nt)entangleme (nostate IR

entangled)(highly state ground True

u

u

u

u

ezd

e

LsKdsiPuIR

Conjecture

K(s) : disentangler, L: scale transformation

(5-4) Emergent Metric from cMERA

We conjecture that the metric in the extra direction is given by using the idea of the quantum metric (up to a constant c):

Note: The quantum distance between two states is defined by

[Nozaki-Ryu-TT, in preparation]

.Vol

)(||)(1

0

2

2

ue d

iLdu

uu

dk

duueucdug

.|1),(2

2121 d

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:

uduuuA egduS

IR

)1(0~

A

B

A

IRuzu log

0UVu

(5-5) Emergent Metric in a (d+1) dim. Free Scalar Theory

Hamiltonian:

Ground state :

Moreover, we introduce the `IR state’ which has no real space entanglement.

)].()()()()([21 22 kkmkkkdkH d

.0ka

.0

).()( 0 i.e.

),()( ,0

A

xx

x

xx

S

xMixMa

xMixMaa

For a free scalar theory, the ground state corresponds to

For the excited states, becomes time-dependent. One might be tempting to guess

Indeed, the previous proposal for guu lead to

|).x|(1-(x) :function offcut a is (x) where

,.).(/)(2

)(ˆ

chaaMkeudkiuK kkud

.)2/1(u) ,0for ( ,/2

1)( 222

2

mMme

es u

u

)(s

222

222 dtgxdedugds tt

u

uuGravity

? |)(| ug ss

Density of bonds

.)( 2uguu

Explicit metric

(i) Massless scalar (E=k)

(ii) Lifshitz scalar (E=kν)

(iii) Massive scalar

222

222 dtgxdedugds tt

u

uuGravity

AdSguu pure the 41

geometry Lifshitz the 4

2

uug

222

2

222

22

2222

4

1

.)/(4

dtgxdmzz

dzds

meeg

tt

u

u

uu

Ｃａｐｐｅｄ　ｏｆｆ　ｉｎ　ｔｈｅ　ＩＲ

(5-6) Excited states after quantum quenches

m(t)

t

.0

021

,0

021

4/1

22

24/1

2

22

4/1

22

24/1

2

22

iktk

iktk

emk

kk

mkB

emk

kk

mkA

m0

).1||||( ,0)( 22 kkkkkk BAaBaA

Note: There is an phase factor ambiguity of (A,B). ⇒ 　 Different choices of time slices ?

Time dependent metric from the Quantum Quench

t

Light cone

looks like a propagation of gravitational wave.

zzgz

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.).

z

[Arrastia-Aparicio-Lopez 10 , Albash-Johnson 10, Balasubramanian-Bernamonti-de Boer-Copland-Craps- Keski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10, 11,….]

(5-7) Towards Holographic Dual of Flat Space

If we consider the (almost) flat metric

the corresponding dispersion relation reads

This leads to the highly non-local Hamiltonian: [cf. Li-TT 10]

,222222 uuu

uu egdxedueds

. 21)( k

ku

ekk

kk eEeE

Eku

u

).()(2

xexdxH d

⑤ Conclusions• The entanglement entropy (EE) is a useful bridge between gravity (string theory) and cond-mat physics.

Gravity Entanglement Cond-mat. systems

• Classical gravity duals + Null energy condition ⇒ a constraint on specific heat 　　⇒ Non-fermi liquids ! 　　　　　　　　　　　　　• Questions: String theory embeddings of the NFL b.g. ? 　　　　　　 　 [See also solutions in Singh 10, Narayan

12]

g AreaAS

.32 with TC

• 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 ? : :