entanglement, conformal field theory, and interfaces · conformal field theory, nucl. phys. b763...
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![Page 1: Entanglement, Conformal Field Theory, and Interfaces · conformal field theory, Nucl. Phys. B763 (2007) 354–430, [hep-th/0607247]. More about relative entropy If the two CFTs are](https://reader034.vdocument.in/reader034/viewer/2022042918/5f5cd2a33d613f0817608a39/html5/thumbnails/1.jpg)
Entanglement,Conformal Field Theory,
and Interfaces
Enrico Brehm • LMU Munich • [email protected]
Ilka Brunner,
Daniel Jaud, Cornelius Schmidt Colinet
with
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Entanglement
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Entanglement
purely quantum phenomenon
non-local “spooky action at a distance”
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Entanglement
purely quantum phenomenon
can reveal new (non-local) aspects of quantum theories
non-local “spooky action at a distance”
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Entanglement
purely quantum phenomenon
can reveal new (non-local) aspects of quantum theories
non-local “spooky action at a distance”
quantum computing
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Entanglement
purely quantum phenomenon
can reveal new (non-local) aspects of quantum theories
non-local “spooky action at a distance”
quantum computing BH physics
monotonicity theoremsAdS/CFTCasini, Huerta 2006
Ryu, Takayanagi 2006
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Entanglement
purely quantum phenomenon
non-local “spooky action at a distance”
observable in experiment (and numerics)
depends on the base and is not conserved
can reveal new (non-local) aspects of quantum theories
quantum computing BH physics
monotonicity theoremsAdS/CFTCasini, Huerta 2006
Ryu, Takayanagi 2006
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Measure of Entanglement
separable state fully entangled state,e.g. Bell state
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Measure of Entanglement
separable state fully entangled state,e.g. Bell state
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Measure of Entanglement
separable state fully entangled state,e.g. Bell stateHow to “order” these states?
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Measure of Entanglement
separable state fully entangled state,e.g. Bell stateHow to “order” these states?
● distillable entanglement● entanglement cost● squashed entanglement
● negativity● logarithmic negativity● robustness
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Measure of Entanglement
separable state fully entangled state,e.g. Bell stateHow to “order” these states?
entanglement entropy
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Entanglement Entropy
AB
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Entanglement Entropy
AB
Replica trick...
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Conformal Field Theory
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Conformal Field Theory
string theory
phase transitions
fixed points in RG
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Conformal Field Theory
QFT invariant under
conformal transformation
holomorphic functions,
Witt algebrain 2 dim.
Virasoro algebra,
c: central charge
quantum
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Conformal Field Theory
QFT invariant under
conformal transformation
holomorphic functions,
Witt algebrain 2 dim.
Virasoro algebra,
c: central charge
2d CFT is organized in
(irred.) reps of Virc
quantum
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Rational Models
finite # of primaries, conformal weights
operator/statecorrespondence
highest weight reps.
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Rational Models
finite # of primaries, conformal weights
operator/statecorrespondence
torus partition function
highest weight reps.
their characters
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Example: Critical Ising Model
at some critical value:
second order phase transition
scale invariance
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Example: Critical Ising Model
at some critical value:
second order phase transition
scale invariance
in continuum limit: free Majorana fermions projected on even fermion numbers
rational model consisting of 3 primaries:
primary conformal weight
id (0,0)
ε (1/2, 1/2)
σ (1/16,1/16 )
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Conformal Interface
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Conformal Interface… or defect
natural generalization of
conformal boundaries
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Conformal Interface… or defect
natural generalization of
conformal boundaries
impurities in quantum chains
String theory:
junction of quantum wires
generalized D-branes?
Stat. mech.:
brane spectrum generatingGraham, Watts 2004
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Conformal Interface… or defect
natural generalization of
conformal boundaries
impurities in quantum chains
String theory:
junction of quantum wires
symmetry generating
generalized D-branes?
Stat. mech.:
RG defects
brane spectrum generatingGraham, Watts 2003
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Conformal Interfaces
operator mapping states from on CFT to the other
Bachas et al 2002
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Conformal Interfaces
operator mapping states from on CFT to the other
gluing condition:
Bachas et al 2002
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Conformal Interfaces
operator mapping states from on CFT to the other
gluing condition:
folding trick
Bachas et al 2002
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Special Gluing Conditions
➢ Both sides vanish independently:
➢ separate boundary conditions
➢ In particular happens when one of the CFTs is trivial
➢ The two components equal independently:
➢ I1,2 also commutes with the Hamiltonian
➢ The interface can be moved around without cost of energy or momentum
➢ This is called a topological interface
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What makes the difference?
boundary changing field
defect changing field
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What makes the difference?
boundary changing field
defect changing field
junction field
generalization
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What else makes the difference?
Fusion of topological defects with other defects or boundaries:
The set of topological
defects form a
(Frobenius) algebraFröhlich et al 2007
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Topological Interfaces in a CFT
acts as a constant map between isomorphic Virasoro representationsPetkova, Zuber 2000
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Topological Interfaces in a CFT
acts as a constant map between isomorphic Virasoro representations
invariance under S-trafo
example: diagonal
rational theories
Petkova, Zuber 2000
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Example: Topological Interfaces of the Ising model
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Conformal Field Theory
Entanglement
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Entanglement Entropy of a Finite Interval
CFT
u v
AB
Cardy, Calabrese 2009
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Entanglement Entropy of a Finite Interval
CFT
u v
AB
remember replica trick:
branch cut
partition function Z(n) on a
complicated Riemann surface
Cardy, Calabrese 2009
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Entanglement Entropy of a Finite Interval
CFT
u v
AB
remember replica trick:
branch cut
partition function Z(n) on a
complicated Riemann surface
CFTn
u v
AB
top. defect “bn”
Cardy, Calabrese 2009
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Entanglement Entropy of a Finite Interval
remember replica trick:
partition function Z(n) on a
complicated Riemann surface
CFTn/bn
u vTn Tn
†
2-point function of twist fields
Cardy, Calabrese 2009
CFT
u v
AB
branch cut
CFTn
u v
AB
top. defect “bn”
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EE of a Finite Interval
2-point function of twist fields
“junction field” of lowest conformal weight
Tn bn
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EE of a Finite Interval
2-point function of twist fields
“junction field” of lowest conformal weight
Tn bn
Cardy condition
S-trafo& leading order
state- operator
correspondence leading order for large L
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Conformal Interfaces
Entanglement
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Entanglement through Conformal Interfaces
CFT
I
branch cut
CFTn
In
bn
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Entanglement through Topological Interfaces
Remember: and
S-trafo & leading order
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Entanglement through Topological Interfaces
Remember: and
S-trafo & leading order no change in the log term
of the EE
contributes to sub-leading term in the EE:
with
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Entanglement through Topological Interfaces
relative entropy / Kullback–Leibler divergence:
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Entanglement through Topological Interfaces
relative entropy / Kullback–Leibler divergence:
diagonal RCFTs
Isin
g
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Entanglement through Non-Topological Interfaces
they affect the leading order contribution
change the conformal weight of the twist field
Example: General interfaces of the Ising model
→ interfaces of the free fermion theory:
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Entanglement through Non-Topological Interfaces
they affect the leading order contribution
change the conformal weight of the twist field
Example: General interfaces of the Ising model
→ interfaces of the free fermion theory:
sep. boundaries
topological
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Entanglement through Non-Topological Interfaces
they affect the leading order contribution
change the conformal weight of the twist field
Example: General interfaces of the Ising model
→projection on even fermion number:
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Entanglement through Non-Topological Interfaces
Sakai, Satoh 2008
transmission coefficient
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Entanglement through Non-Topological Interfaces
they affect the leading order contribution
change the conformal weight of the twist field
Some interesting questions:
➢ How does the EE behave for general non-topological defects?
➢ On which features of a general conformal defect does it depend? Keywords: transmission coefficient; Casimir energy; topological data.
➢ Is the sub-leading term always constant under non-topological deformations of a topological defect?
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Final Words and Thoughts
➢ By unfolding a boundary one may interpret it as a top. defect in a chiral theory
➢ one can use the same techniques to derive the left-right entanglement at a boundary
➢ The entanglement through the defect is a feature of the defect itself.
➢ It might be possible to define more structure to the space of 2d CFTs
➢ define distances between CFTs, by the help of conformal defects and the EE through them? (in the spirit of ideas of Bachas et al 2014 )
➢ the infinitesimal limit of the Kullback–Leibler divergence yields the Fisher information metric
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Thank You!
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H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B600 (2004) 142 [hep-th/0405111].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073].
K. Graham and G. M. T. Watts, Defect lines and boundary flows, JHEP 04 (2004)019, [hep-th/0306167].
V. B. Petkova and J. B. Zuber, Generalized twisted partition functions, Phys. Lett. B504(2001) 157–164, [hep-th/0011021].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J.Phys. A42 (2009) 504005, [arXiv:0905.4013].
K. Sakai and Y. Satoh, Entanglement through conformal interfaces, JHEP 0812 (2008) 001, [arXiv:0809.4548].
C. P. Bachas, I. Brunner, M. R. Douglas, and L. Rastelli, Calabi’s diastasis as interfaceentropy, Phys. Rev. D90 (2014), no. 4 045004, [arXiv:1311.2202].
C. Bachas, J. de Boer, R. Dijkgraaf, and H. Ooguri, Permeable conformal walls and holography, JHEP 06 (2002) 027, [hep-th/0111210].
J. Fröhlich, J. Fuchs, I. Runkel, and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B763 (2007) 354–430, [hep-th/0607247].
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More about relative entropy
If the two CFTs are not the same: Their exists a defect s.t. the Kullback-Leibler divergence vanishes iff the spectra are identical.
Using the constrains for dAi:
so they form a probability distribution.
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Results for higher torus models
Bachas et al 2012
is also the g-factor of the interface