entanglement in quantum critical phenomena, holography and gravity
DESCRIPTION
Entanglement in Quantum Critical Phenomena, Holography and Gravity. Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA. hep-th/0602134 hep-th/0606184. Banff, July 31, 2006. gravity - quantum information -condensed matter. finding entanglement entropy in spin chains - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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)
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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
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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”)
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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
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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
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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|
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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
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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
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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”
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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
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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!
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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!
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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)
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gravitational coupling
1 22N
m mF G
r - gravitational force between two
bodies
NG is determined by the microscopical properties of a fundamental theory
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● 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
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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
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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
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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 :
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● 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
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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
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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
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the holographic formula enables one to
compute entanglement entropy in strongly
coupled theories by using geometrical
methods
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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
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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)
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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
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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)
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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
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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