Holographic Entanglement Entropy of Excited States Tadashi - - PowerPoint PPT Presentation

Gauge/Gravity Duality 2013@Max Plank Inst. , Munich July 29th –August 2nd ,2013

Holographic Entanglement Entropy of Excited States

Tadashi Takayanagi (YITP, Kyoto Univ.)

Based on arXiv:1212.1164 (PRL 110, 091602 (2013)) with Jyotirmoy Bhattacharya (Kavli IPMU), Masahiro Nozaki (YITP), Tomonori Ugajin (Kavli IPMU, YITP) arXiv:1302.5703 (JHEP 05(2013)080) with Masahiro Nozaki (YITP), Tokiro Numasawa (YITP) arXiv:1304.7100 with Masahiro Nozaki (YITP), Tokiro Numasawa (YITP), Andrea Prudenziati (YITP) + work in progress with Jyotirmoy Bhattacharya (Kavli IPMU)

① Introduction

What is the entanglement entropy (EE) ?

A measure how much a given quantum state is quantum mechanically entangled (or complicated). ~ The amount of `active’ degrees of freedom (or its information)

Why interesting and useful ?

At present, it seems still difficult to observe EE in real experiments (→ a developing subject). But, recently it is very common to calculate EE in `numerical experiments’ of cond-mat systems. e.g. computing central charges, detecting spin liquids

Advantages of EE

• EE = A quantum order parameter (~a generalization
• f `Wilson loops’) Classify quantum phases.
• The entanglement entropy (EE) is a helpful bridge

between gravity (string) and cond-mat physics. Gravity Entanglement Cond-mat. systems

• A universal quantity which characterizes the

properties of non-equilibrium states.



g Area 

A

S 

AdS/CFT (Holography)

Quantum Many-body Systems (Cond-mat, QFTs, CFTs, …..) Quantum gravity String theory Quantum Information Theory AdS/CFT (Holography) HEE, BH info. EE, ES, Tensor networks, etc.

Information vs. Energy

1st law of thermodynamics: T ・

dS = dE

• Temp. Information Energy

⇒ Can we find an analogous relation in any quantum systems which are far from the equilibrium ? Something like: Tent ・

dSA = dEA

?? Information in A Energy in A = EE Can we observe EE ?? The main motivation of this talk. What ?

Contents

① Introduction ② Basic Facts about the Entanglement Entropy (EE) ③ `The First law’ for the EE of Excited States ④ Entanglement Density and SSA ⑤ Holographic Local Quenches and EE ⑥ What is the Einstein equation for HEE ? ⑦ Conclusions

Divide a quantum system into two subsystems A and B. Define the reduced density matrix for A by taking a trace over the Hilbert space of B . Now the entanglement entropy is defined by the von-Neumann entropy:

A



.

B A tot

H H H  

A B Example: Spin Chain

② Basic Facts about the Entanglement Entropy (EE)

(2-1) Definition of Entanglement Entropy

B A   

B

A

slice time : N

(2-2) Basic Properties of EE (i) If is a pure state (i.e. ) and then (ii) Strong Subadditivity (SSA) [Lieb-Ruskai 73] When for any ,

,

B A tot

H H H  

⇒ EE is not extensive !

,

C B A tot

H H H H   

,

B C B A C B B A

S S S S   

   

.

C A C B B A

S S S S   

 

B A C (Actually, these two inequalities are equivalent .)

The strong subadditivity can also be regarded as the concavity of von-Neumann entropy. Indeed, if we assume A,B,C are numbers, then

x)

• f

function concave (i.e. . ) ( ), ( ) ( 2 2 ), ( ) ( ) ( ) (

2 2

                    x S dx d y S x S y x S B S C B A S C B S B A S

x S(x)

(2-3) Area law EE in QFTs includes UV divergences.

In a d+1 dim. QFT (d>1) with a UV relativistic fixed point, the leading term of EE at its ground state behaves like where is a UV cutoff (i.e. lattice spacing). [d=1: log div.] Intuitively, this property is understood like: Most strongly entangled

terms), subleading ( A) Area( ~

1

 

 d A

a S

a

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

A ∂A Area Law

(2-4) Holographic Entanglement Entropy (HEE)

[Ryu-TT 06]

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

homologous

Note: In time-dependent b.g., we need to employ the covariant version [Hubeny-Rangamani-TT 07].

. ~ and

A A

A A     

2 d

AdS 

z

A

) direction. time

• mit the

(We

B

A



1 d

CFT 

2 1 2 2 2 2 2

z dz dx dt R ds

d i i AdS AdS

 

   

• ff)

cut (UV   z 

A



Verification of HEE

• Confirmations of basic properties:

Area law, Strong subadditivity (SSA), Conformal anomaly,….

• Direct Derivation of HEE from AdS/CFT:

(i) Pure AdS, A = a round sphere [Casini-Huerta-Myers 11] (ii) Euclidean AdS/CFT [Lewkowycz-Maldacena 13, cf. Fursaev 06] (iii) Disjoint Subsystems [Headrick 10, Faulkner 13, Hartman 13] (iv) General time-dependent AdS/CFT → Not yet.

[But, ∃confirmations of SSA: Allais-Tonni 11, Callan-He-Headrick 12, Wall 13]

• Corrections to HEE beyond the supergravity limit:

[Higher derivatives: Hung-Myers-Smolkin 11, de Boer-Kulaxizi-Parnachev 11,….. ] [1/N effect: Barrella-Dong-Hartnoll-Martin 13, Faulkner-Lewkowycz-Maldacena 13] [Higher spin gravity: de Boer-Jottar 13, Ammon-Castro-Iqbal 13]

Holographic Proof of Strong Subadditivity The holographic proof of SSA inequality is very quick !

B C B A C B B A

S S S S    

   

A B C

=

A B C



A B C

C A C B B A

S S S S    

 

A B C

=

A B C



A B C

[Headrick-TT 07]

General Behavior of HEE

[Ryu-TT 06]

divergence law Area

Agrees with conformal anomaly (central charge)

[Calabrese-Cardy 04, Solodukhin 08, Hung-Myers-Smolkin 11 …]

A universal quantity F which characterizes odd dim. CFT. ⇒A proof of c-theorem in 3 dim. (F-theorem). [Casini-Huerta 12, Liu-

Mezei 12, Myers-Singh 12, …]

A

③ `The First law’ for EE of Excited States

[Bhattacharya-Nozaki-Ugajin-TT 12]

(3-1) Outline

Since the EE in a QFT is UV divergent, we would like to focus on the difference between the values of EE. In other words, we will consider excited states and calculate: This is always finite and we will compare this entropy with the energy in A:

.

State Ground A A A

S S S   

.



 

A tt d A

T dx E

(3-2) Holographic Calculation

Consider an asymptotically AdSd+2 background (= an excited state in CFTd+1):

 

. 16 ..... 1 ) ( ..., 1 ) ( , ) ( ) (

1 1 1 1 2 2 2 2 2 2 N d tt d d d i i

G m dR T mz z g mz z f dx dz z g dt z f z R ds  

   

            



z ∞

AdS bdy

UV IR ???

We do not care the details of IR.

energy density

Holographic Entanglement Entropy Analysis If we assume a small subsystem A with the size such that then we can show The `entanglement temperature’ is given by The constant c is universal in that it only depends on the shape

• f the subsystem A:

, 1

1   d

ml

l

. , where ,

AdS Pure



        

A tt d A A A A A A ent

T dx E S S S E S T . l c Tent 

sphere. round a A when 2 2 . .     d c g e

A

Holographic Prediction Consider an exited state in a CFT which has an approximate translational and rotational invariance. If the size of the subsystem A (= ) is small enough such that then the following `1st law’ like relation is satisfied: Note 1: The constant c depends only on the geometry of A. Note 2: For more general critical points with the dynamical exponent z, we have

), ( /

2 1

N O G R l T

N d d tt

  



l

, , l c T E S T

ent A A ent

    

.

z ent

l c T



  Energy Info.

Example 1. Excited States in 2d CFT Example 2. 3d CFT at finite temp.

. 3 ) ( 3 2

2 2 2 A tot A

E l l l h h S        

system the

• f

length total the dim. conformal ) , (  

tot

l h h

[Agrees with Alcaraz-Berganza-Sierra 11]

. ) ( ) ( , ) , ( ) , ( ) , (

1

            

 

l T T l l c T l T S d l T E d l T T

BH ent ent fixed l BH A BH A BH ent

0.05 0.10 0.15 0.20 0.25 0.30 l 2 4 6 8 10 12 T

TBH=3 TBH=2 TBH=1

[Nozaki-Numasawa-TT 13]

We focus on the EE for a pure state in 2d CFTs for simplicity. Let us estimate the EE for the subsystem A (=an interval) by summing all of the EE between two infinitesimal regions:

. ) , ( ) , ( ) , (

2 1 1 2

2 1

  

       

   l l l l A

y x n dy x y n dy dx l l S

x l1

y n(y,x) n(x,y)

l2 y

A

④ Entanglement Density

. ) , ( ) , (

2 1 1 2

  

       

   l l l l A

y x n dy x y n dy dx S

). , ( ) , ( ) , (

2 1 1 2

2 2 2 2

  

    

   l l l l A

l x dxn y l n dy l y n dy l S

Therefore we find We will call the entanglement density. Clearly this quantity should be non-negative. As we will see, this fact comes from the SSA.

). , ( 2

2 1 2 1 2

l l n l l S A     ) , (

2 1 l

l n

Now, let us apply the SSA relation to the intervals:

. ) , ( , ) , ( ), , ( ) , ( ) , ( ) , ( ,

2 3 2 4 1 4 2 3 1

             

   

y x n y x y x S l l S l l S l l S l l S S S S S

B C B A C B B A

l1 l2 l3 l4

A B C

Note: This property is true for any excited states.

• Cf. Entropic C-theorem

[Casini-Huerta 04]

Consider a ground state of a QFT. (i.e.∃Lorentz invariance) We have We set

t x

Light cone

The 1st law and entanglement density In 2d CFTs, we find the following property from the 1st law: In terms of the entanglement density, we obtain:

conserved. ) , , ( lim ), ( ) , ( 3 ) , , (

3 2

     



   

t l S d l O t T l t l S

A l tt A

    

conserved. ) , , ( ). , ( 3 ) , , ( lim      



   

t n d t T t l n

tt l

    

. 2 / ) ( ,

2 1 1 2

l l l l l     

Entanglement ⇔ Energy

Full conservation law of entanglement density [Proof] We compactify the space on a circle with periodicity L.

Then we find

. ) , , ( ) , , ( : Claim

2 1 2 1 2 1

 

    t l l n dld t l l n dl dl 

). , , ( ) , , ( t L l L n t l n S S

B A

        

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

1 1 1 2 1 2 1 2 2 / 2 1 2 /

1 1 2

                     

    

  L l x A A A L L l l L L

l x S x L l x S x dl l l l l S dl dl t l n dl d  

=0 because SA=SB. =0 because 1st law.

⑤ Holographic Local Quenches and EE

[Nozaki-Numasawa-TT 13]

(5-1) Outline

We would like to study the relation between the EE and energy when we excite the system locally.

A

Localized Excitations

A

S   Calculate

~The amount of quantum Information of excitations

(5-2) Local Quantum Quenches

Global Quench (Global Excitations) The thermalization under a sudden change of Hamiltonian is called quantum quench and has been intensively studied in condensed matter physics. [Calabrese-Cardy 05-10] This corresponds to the time-dependent setup of AdS/CFT: Black hole formation in AdS ⇔ Thermalization in CFT

Local Quenches (Local excitations) Original Local Quench: Generalization (Local Excitations): We consider this kind of example using the AdS/CFT.

(5-3) Holographic Local Quenches

We argue that a simple model of holographic local quench is given by a free falling particle (mass m) in AdSd+2.

2 2 2 2 2 2

z x d dt dz R ds     

Trajectory:

α~ the size of localized excitations

. ) (

2 2

   t t z

Construction of Back-reacted Solutions We use the method first noticed by [Horowitz-Itzhaki 99]. Start with the global AdS BH:

. / ) ( , ) ( ) (

2 2 2 2 2 2 2 2 2 

       

d d

r M R r r f d r dr r f R d r f ds 

. 2 ) ( ), ,..., 2 , 1 ( , sin , 2 ) ( cos

2 2 2 2 1 2 2 2 2 2 2 2 2

z t x z e e R r d i z Rx r z Rt r R z t x z e e R r R

d i i

                 

      

 

Coordinate transformation

Asymptotic AdS space (M=0 → Pure AdS) Z

AdS boundary

Energy density via AdS/CFT

d=2 d=3 d=4

t z

Analysis of HEE

(i) Perturbation theory w.r.t M → Any dim. d. → We can confirm the `1st law’ in the small size limit. (ii) Exact analysis → So far only for AdS3 (d=1).

. ) 1 ( ) 2 / ( 8 ,

1 2 / 2 

     

d N particle

d d R mG M m R E  

Basic picture

Time Evolution of HEE in AdS3/CFT2 (Case 1)

M=0.1, R=1, l=5 M=5, R=1, l=5

A

S

A

S t t l

Time Evolution of HEE in AdS3/CFT2 (Case 2)

A

S

A

S t t

M=0.01, R=1, l=5 M=500, R=1, l=5

l

If we take the limit l>>t>>α in the case 2, we find

. const log 3 log 6    a l c t c S A 

Subsystem A

. const log 6 log 3 procedure joint by the quenches local for result CFT 2d .     l c a t c S cf

A

α

Entanglement Density

) , (

2 1 l

l n

1

l

2

l

α

t=0

(5-4) Quantum Information and Energy We can estimate the maximal value of when we varies l and t. For a small M, we find for any dim. 1<<Δ<<c

A

S 

. e ~ s Excitation Localized

• f

s Microstate # ) 4 , 2 / , 2 ( .

s

C 3 2 1 max  

          C C C C mR C S

d d



Exact result in AdS3 Summary:

The amount of quantum information

• f Localized excitations ~ E ・ α

(`fire ball’ of gluons)

. 2 2 sin log 3

2 2 max

                      R M R M R R c S 

α

Energy Size

• Cf. Bekenstein Bound

⑥ What is the Einstein equation for HEE ?

Gravity on AdS CFT Einstein eq. What ?

) , , ( l x t SA 

AdS/CFT The `1st law-like’ relation appears only when the size of A is small. What can we say if the size of subsystem is not small ? This is related to a basic question in the AdS/CFT: Below we study a HEE counterpart of perturbative Einstein eq. assuming small excitations of a CFT.

[Nozaki-Numasawa-Prudenziati-TT 13, Bhattacharya-TT in preparation]

AdS4/CFT3

Let A be a round ball with radius l. Its center is situated at . The perturbative Einstein equation is rewritten as follows Note: There are no time derivatives. ⇒ This gives a constraint for HEE at a fixed time. The time evolution is determined by IR bdy conditions.

O O l x t S l

A x l l

              ) , , ( 3

2 2 2





Matter field contributions C.C. Kinetic term

O   ) , ( x t 

AdS4 Schwarzschild BH

When the size of the subsystem is very large , we find This coincides with the holographic result for flat space.

AdS3/CFT2

In AdS3 gravity, we have two constraints:

[Confirmed in CFT2 by Wong-Klich-Pando Zayas-Vaman 13]

x-l x+l

A

 

. ) , , ( 2 , ) , , (

2 2 2 2 2

O O l x t S l O O l x t S

A t l A t x

                

 

. ) , , (

2 2

O O l x t SA

x l l

       



Intuitive interpretation of these constraints

Hyperbolic PDE: 



|). | ( |) | ( ) , , (

2 2

x l g x l f S l x t S

A A x l

           



Local excitations

x

A

S 

becomes non-trivial only when ∂A intersects with the excited region

. | | x l   

⑦ Conclusions

• The amount of quantum information ~ energy in CFTs.

→ Analogous to the thermodynamical first law.

• A simple holographic model of local quenches and its HEE.
• A counterpart of Einstein equation for entanglement entropy.

→ Can we employ these to study the holography in other spacetimes (e.g. de-Sitter spaces) ? → Can we formulate quantum gravity in terms of HEE ?