Entanglement in Quantum Critical Phenomena, Holography and Gravity Dmitri V. Fursaev Joint Institute...
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Transcript of Entanglement in Quantum Critical Phenomena, Holography and Gravity Dmitri V. Fursaev Joint Institute...
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