Entanglement, Conformal Field Theory, and...

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Entanglement, Conformal Field Theory, and Interfaces Enrico Brehm • LMU Munich • [email protected] Ilka Brunner, Daniel Jaud, Cornelius Schmidt Colinet with

Transcript of Entanglement, Conformal Field Theory, and...

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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