Entanglement in Quantum Critical Phenomena, Holography and Gravity

Dmitri V. Fursaev

Joint Institute for Nuclear Research

Dubna, RUSSIA

Banff, July 31, 2006

hep-th/0602134hep-th/0606184

gravity - quantum information -condensed matter

finding entanglement entropy in spin chains

near a critical point

finding a minimal surface in a curved space

one dimension higher

plan of the talk

● quantum entanglement in 2D critical phenomena and CFT’s

● geometrical structure of entanglement entropy

● gravitational coupling in quantum gravity and entanglement entropy

● new gravity analogs in condensed matter systems (applications)

● “holographic formula” for entanglement entropy (in QFT’s dual to AdS gravity)

Quantum Entanglement

Quantum state of particle «1» cannot be described independently from particle «2» (even for spatial separation at long distances)

1 2 1 2

1| (| | | | )

2

measure of entanglement

2 2 2

2 1

( ln )

(| |)

S Tr

Tr

- entropy ofentanglement

density matrix of particle «2» under integration over the states of «1»

«2» is in a mixed state when information about «1» is not availableS – measures the loss of information about “1” (or “2”)

Ising spin chains

11

( )N

X X ZK K K

K

H

2

1( , ) log

6 2

NS N

2

1( , ) log | 1|

6S N

1 | 1| 1 off-critical regime at large N

critical regime 1

RG-evolution of the entropy

entropy does not increase under RG-flow (as a result of integration of high energy modes)

IR IR

UV

1 is UV fixed point

Explanation

Near the critical point the Ising model is equivalent to a 2D quantum

field theory with mass m proportional to

At the critical point it is equivalent to a 2D CFT with 2 massless

fermions each having the central charge 1/2

| 1|

Entanglement in 2D models:analytical results

1ln

6

cS

ma

1ln6

LcS

a

1L

1ln sin 26

Lc LS g

a L

L

11/ m La is a UV cutoff

Calabrese, Cardyhep-th/0405152

ground state entanglementon an interval

massive case:

massless case:

is the length of

analytical results (continued)

1ln sin3

Lc LS

a L

1ln sinh3

LcS

a

1/T

ground state entanglement for asystem on a circle

system at a finite temperature

1L is the length of

effective action and geometrical structure of entanglement entropy

-effective action is defined on manifolds with cone-like singularities

- “inverse temperature”

1 1 1 2

1 2

( ) lim lim 1 ln ( , )

( , )

ln ( , )

2

nnS T Tr Z T

n

Z T Tr

Z T

n

- “partition function”

example: 2D theory at a finite

temperature T

3n

/1 2

H TTr e

31 1Tr case

conical singularity is located at the separating point

( 2 , )

( )

Z T

Z T

- standardpartition function

effective action on a manifold with conical singularities is the gravity action

(even if the manifold is locally flat)

curvature at the singularity is non-trivial:

(2)2(2 ) ( )R B

derivation of entanglement entropy in a flat space has to do with gravity effects!

many-body systems in higher dimensions

a

spin lattice continuum limit

2

AS

a A – area of a flat separation surface which divides

the system into two parts (pure quantum states!)

entropy per unit area in a QFT is determined by a UV cutoff!

geometrical structure of the entropy

2ln

A LS C a

a a

edge (L = number of edges)

separating surface (of area A)

sharp corner (C = number of corners)

(method of derivation: spectral geometry)

(Fursaev, hep-th/0602134)

for ground statea is a cutoff

C – topological term (first pointed out in D=3 by Preskill and Kitaev)

gravitational coupling

1 22N

m mF G

r - gravitational force between two

bodies

NG is determined by the microscopical properties of a fundamental theory

● gravitational constant as a measure of quantum entanglement in the

fundamental theory

3

4FUNDN

cs

G

FUNDs - entanglement entropy per unit area for degrees of freedom of the fundamental theory in a flat space

CONJECTURE (Fursaev, hep-th/0602134)

( 4)d

arguments:

● entropy density is determined by UV-cutoff

● the conjecture is valid for area density of the entropy of black holes

● entanglement entropy can be derived form the effective gravity action

● entropy in QFT’s which admit AdS duals

BLACK HOLE THERMODYNAMICS

3

4BH H

N

AS c

G

HA

Bekenstein-Hawking entropy

- area of the horizon

BHS - measure of the loss of information about states underthe horizon

some references: ● black hole entropy as the entropy of entanglement (Srednicki 93, Sorkin et

al 86)

● iduced gravity (Sakharov 68) as a condition (Jacobson 94, Frolov, Fursaev, Zelnikov 96)

● application to de Sitter horizon (Hawking, Maldacena, Strominger 00)

● entropy of certain type black holes in string theory as the entanglement entropy in 2- and 3- qubit systems (Duff 06, Kallosh & Linde 06)

● yields the value for the fundamental entropy in flat space in terms of gravity coupling

● horizon entropy is a particular case

our conjecture :

● applications: new gravity analogs in condensed matter systems

14

EFF

sG

s

In condensed matter systems one can define an effective gravity constant

where is the ground state entanglement entropy per unit area

Requirements:

● lattice models (cutoff)● second order phase transition● description in terms of a massive QFT near the critical point

Advantage: one does not need to introduce effective metric in the system

( 1)c

theories with extra dimensions

(4 )

1

4FUND ns

G

(4 )nG

n

the conjecture should hold in higher dimensions: fundamental entanglement entropy per unit area of the separating surface is

is the higher-dimensional gravitational coupling

What is the separating surface in higher dimensions?

● Kaluza-Klein-like theories:

● brane-world models(only gravity is higherdimensional):

- space of extra dimensions

extension of the separating surface to higher dimensions has to be determined by the dynamical gravity equations in the bulk

nis

Holographic Formula for the Entropy

A

( 1)4 d

AS

G

Ryu and Takayanagi,hep-th/0603001, 0605073

CFT which admit a dual description in terms of the Anti-de Sitter (AdS) gravity one dimension higher

( 1)dG

Let be the extension of the separating surface in d-dim. CFT

1) is a minimal surface in (d+1) dimensional AdS space

2) “holographic formula” holds: is the area of

is the gravity couplingin AdS

the holographic formula enables one to

compute entanglement entropy in strongly

coupled theories by using geometrical

methods

example in d=2:CFT on a circle

0

0

0

2 2 2 2 2 2 2

2 211

2 2 10

1

3

3

cosh sinh

2

cosh 1 2sinh sin

ln sin4 3

3

2

CFT

ds l d dt d

l

Lds ds

LLA

l L

Le

a

LA cS e

G L

lc

G

- AdS radius

A is the length of the geodesic in AdS

- UV cutoff

-holographic formula reproducesthe entropy for a ground stateentanglement

- central charge in d=2 CFT

Sketch of the proof of the holographic formula

2

[ ]

23 3

( ) lim 1 ln ( , )

2

( , ) [ ]

1 1 1[ ] , 2

16 8n n

I g

M M

n n

S T Z T

n

Z T Dg e

I g R K dG l G

M M

Fursaev, hep-th/0606184

-AdS/CFT representation for CFT partition function (with specific boundary conditions)

is (a conformal) boundary of

(3D AdS / 2D CFT)

the proof (continued)

2

3

3

( ) lim 1 [ ]

2(2 )

[ ] [ ] (2 )8

[ ] 0 0

4

regular

regular

S T I g

R R A

AI g I g

G

I g A

AS

G

in semiclassical approximation

extremality of the action requiresbe a minimal surface

there are conical singularities in the bulk located on

consequences

• possibility to consider entropy in stationary but not static theories (Riemannian sections)

• choice of the minimal surface in case of several options

• theories with different phases and phase transitions

• higher-curvature corrections in the bulk

• entropy in brane-world models (Randall and Sundrum)

choice of the minimal surface infinite-temperature cases and topology

Euclidean BTZ black hole slice of the torus

The bulk manifold is obtained by cutting and gluing alongn copies of the torus

Summary

- Entanglement and critical phenomena in condensed matter systems (d=2,...)

- Entanglement in quantum gravity: relation to gravity coupling in a fundamental theory

- New gravity analogs in condensed matter (lattice models)

- “Holographic” representation of entanglement entropy: geometrical way of computation + new ideas