Review of Holographic (and other) computations of...

Holographic(and other)

computations of

Review of

computations of Entanglement Entropy, and its role in gravity

Entanglement Entropy

• in QFT, typically introduce a (smooth) boundary or entanglingsurface which divides the space into two separate regions

• integrate out degrees of freedom in “outside” region• remaining dof are described by a density matrix

• what is entanglement entropy?general tool; divide quantum system into two parts and useentropy as measure of correlations between subsystems

AB

calculate von Neumann entropy:


AB

• remaining dof are described by a density matrix


R

• result is UV divergent! dominated by short-distance correlations

= spacetime dimension

= short-distance cut-off• must regulate calculation:

• careful analysis reveals geometric structure, eg,


AB

• remaining dof are described by a density matrix


R

= spacetime dimension

= short-distance cut-off• must regulate calculation:

• leading coefficients sensitive to details of regulator, eg, • find universal information characterizing underlying QFT insubleading terms, eg,

General comments on Entanglement Entropy:

• nonlocal quantity which is (at best) very difficult to measure

• in condensed matter theory: diagnostic to characterize quantumcritical points, quantum spin fluids or topological phases

• in quantum information theory: useful measure of quantum

no accepted experimental procedure

• in quantum information theory: useful measure of quantumentanglement (a computational resource)

Where did “Entanglement Entropy” come from? black holes

• recall that leading term obeys “area law”:

suggestive of Bekenstein-Hawking formula if

(Sorkin `84; Bombelli, Koul, Lee & Sorkin; Srednicki; Frolov & Novikov; . . .)

• Sorkin ’84: looking for origin of black hole entropy

General comments on Entanglement Entropy:

(Sorkin `84; Bombelli, Koul, Lee & Sorkin; Srednicki; Frolov & Novikov; . . .)

• problem?: leading singularity not universal; regulator dependent

• active topic in 90’s but story left unresolved (return to this later)

• recently considered in AdS/CFT correspondence(Ryu & Takayanagi `06)

AdS/CFT Correspondence:

Bulk: gravity with negative Λin d+1 dimensions

anti-de Sitterspace

Boundary: quantum field theoryin d dimensions

conformalfield theory

“holography”

energyradius

Holographic Entanglement Entropy:(Ryu & Takayanagi `06)

AB

AdS boundary

AdS bulk

boundaryconformal field

theory

gravitationalpotential/redshiftAdS bulk

spacetime

potential/redshift

• “UV divergence” because area integral extends to

Holographic Entanglement Entropy:

AB

AdS boundary

(Ryu & Takayanagi `06)

cut-off in boundary CFT:

cut-off surface

• finite result by stopping radial integral at large radius:

short-distance cut-off in boundary theory:

• “UV divergence” because area integral extends to


AB

AdS boundary



central charge(counts dof) “Area Law”

cut-off surface


AB

AdS boundary



cut-off surface

general expression (as desired):

(d even)

(d odd)universal contributions


conjecture

Extensive consistency tests:


1) leading contribution yields “area law”

2) recover known results of Calabrese & Cardyfor d=2 CFT

(also result for thermal ensemble)


conjecture



3) in a pure state

and both yield same bulk surface V

AdSboundary



conjecture



3) in a pure state


AdSboundary


cf: thermal ensemble ≠ pure state horizon in bulk

AdS/CFT Dictionary: black holethermal bath


conjecture



4) Entropy of eternal black hole =entanglement entropy of boundary CFT & thermofield double

(Headrick)(Headrick)

boundaryCFT

thermofielddouble

extremal surface =bifurcation surface


conjecture



5) sub-additivity:(Headrick & Takayanagi)

[ “all” other inequalities: Hayden, Headrick & Maloney]


conjecture



4) Entropy of eternal black hole =entanglement entropy of boundary CFT & thermofield double

5) sub-additivity:(Headrick & Takayanagi)

for more general holographic framework, expect

includes correctionscorrections

(Hayden, Headrick & Maloney)

some progress with classical higher curvature gravity:

for more general holographic framework, expect

• note is not unique! and is wrong choice!

• correct choice understood for “Lovelock theories”

(Hung, Myers & Smolkin)(deBoer, Kulaxizi & Parnachev)

(Solodukhin)• test with universal term for d=4 CFT: (Solodukhin)

thermal entropy

universal contribution

c a

• test with universal term for d=4 CFT:

Topics currently trending in Holographic SEE:

• suggest C/F-theorems

• diagnostic of holo-fermi surfaces

• probe of holo-quantum quenches

• probe of holo-RG flows

(Myers & Sinha; Liu & Mezei; . . .)

(Ogawa, Takayanagi & Ugajin; Huijse, Sachdev & Swingle; . . . )

(Albash & Johnson; Myers & Singh; . . .)

(Abajo-Arrastia, Aparicio, Lopez; Balasubramanian et al; . . .)

• probe of holo-RG flows

• probe of holo-insulator/superconductor transition

• probe of holo-QFT in curved/deSitter space

• probe of holo-logarithmic CFT’s

• probe of holo-fractionalized states

(Albash & Johnson; Myers & Singh; . . .)

● ● ● ●

(Grumiller, Rieldler, Rosseel & Zojer)

(Maldacena & Pimental)

(Albash, Johnson & Macdonald; Hartnoll & Huijse, . . .)

(Albash & Johnson; Cai, He, Li & Li . . .)

????

Lessons for bulk gravity theory:

AdS/CFT Dictionary:

Bulk: black holeBoundary: thermal plasma

Temperature Temperature

Energy Energy

Entropy Entropy


(entanglement entropy)boundary

= (entropy associated with extremal surface)bulk

What Entropy?/Entropy of what?

extremalsurface

What are the rules?

(eg, Hubney, Rangamani & Takayanagi)

not black hole! not horizon!not causal domain!

Proposal : Geometric Entropy• in a theory of quantum gravity, for any sufficiently large regionwith a smooth boundary in a smooth background (eg, flat space),leading contribution to entanglement entropy takes the form:

(Bianchi & Myers)

AB

• in QG, (short range) quantum entanglement corresponding to area law is a signature of macroscopic spacetime geometry

(entanglement entropy)boundary

= (entropy associated with extremal surface)bulk

not black hole! not horizon!not causal domain!

• R&T construction assigns entropy to bulkregions with “unconventional” boundaries:


not causal domain!

extremalsurface

• indicates S BH applies more broadly!

• what about extremization?

needed to make match above

• other surfaces might be expected to giveother entropic measures of entanglementin boundary theory(Hubeny & Rangamani; Balasubramanian, McDermott & van Raamsdonk)

Proposal : Geometric Entropy• in a theory of quantum gravity, for any sufficiently large regionwith a smooth boundary in a smooth background (eg, flat space),there is an entanglement entropy which takes the form:

• evidence comes from several directions:

(Bianchi & Myers)


1. holographic in AdS/CFT correspondence

2. QFT renormalization of

3. Randall-Sundrum 2 model

4. spin-foam approach to quantum gravity

5. Jacobson’s “thermal origin” of gravity

Where did “Entanglement Entropy” come from? black holes

• recall that leading term obeys “area law”:

suggestive of BH formula if (Sorkin `84; Bombelli, Koul, Lee & Sorkin; Srednicki; Frolov & Novikov)

• Sorkin ’84: looking for origin of black hole entropy

• problem?: leading singularity not universal; regulator dependent• problem?: leading singularity not universal; regulator dependent

• resolution: this singularity represents contribution of “low energy”d.o.f. which actually renormalizes “bare” area term

(Susskind & Uglum)

both coefficients are regulator dependentbut for a given regulator should match!

BH Entropy ~ Entanglement Entropy (Susskind & Uglum)


• seems matching is not always working??

• “a beautiful killed by ugly facts”??

• seems matching is not always working??(Demers, Lafrance & Myers; Kabat, Strassler, Frolov, Solodukhin, Fursaev, . . . )

• all numerical factors seem to be resolved! (Cooperman & Luty)

matching of area term works for any QFT (s=0,1/2, 1, 3/2)to all orders in perturbation theory for any Killing horizon

• technical difficulties for spin-2 graviton

• results apply for Rindler horizon in flat space♥(some technical/conceptual issues may remain! Jacobson & Satz)

BH Entropy ~ Entanglement Entropy (Susskind & Uglum)


• “a beautiful killed by ugly facts”??

• result “unsatisfying”: • result “unsatisfying”:

where did bare term come from?

• formally “off-shell” method is precisely calculation of SEE

(Susskind & Uglum; Callan & Wilczek; Myers & Sinha: extends to Swald)

• challenge: understand microscopic d.o.f. of quantum gravityAdS/CFT: eternal black hole (or any Killing horizon)

(Maldacena; van Raamsdonk et al; Casini, Huerta & Myers)

AB

• first step in calculation of SEE is to determine

• return to considerations of SEE of general region in some QFT

• encodes standard correlators, eg, if global vacuum:

• by causality, describes physics throughout causal domain

AB

Entanglement Hamiltonian:

• hermitian and positive semi-definite, hence

= modular or entanglement Hamiltonian

• formally can consider evolution by

• unfortunately is nonlocal and flow is nonlocal/not geometric

AB

Entanglement Hamiltonian:

• hermitian and positive semi-definite, hence

• explicitly known in only a few instances

• most famous example: Rindler wedge

ABboost generator●

= modular or entanglement Hamiltonian

A

Entanglement Hamiltonian:Can this formalism provide

insight for “area law” term in S EE?

• zoom in on infinitesimal patch of

region looks like flat spacelooks like Rindler horizon

• assume Hadamard-like statecorrelators have standard UV sing’s

• UV part of same as in flat space• must have Rindler term:

• must have Rindler term:

A



• UV part of must besame as in flat space

for each infinitesimal patch:

• must have Rindler term:

hence SEE must contain divergent area law contribution!

• invoke Cooperman & Luty: area law divergence matchesprecisely renormalization of :

for any large region of smooth geometry!!

Rindler yields area law; hence

A



• challenge: understand microscopic d.o.f. of quantum gravity

• we can formulate a formal geometric argument analogous tothe “off-shell” method for black holes (or Rindler horizons)

• Cooperman & Luty: area law divergencematches precisely renormalization of :

for any large region of smooth geometry!!

Proposal : Geometric Entropy• in a theory of quantum gravity, for any sufficiently large regionwith a smooth boundary in a smooth background (eg, flat space),there is an entanglement entropy which takes the form:


(Bianchi & Myers)


1. holographic in AdS/CFT correspondence

2. QFT renormalization of

3. Randall-Sundrum 2 model

4. spin-foam approach to quantum gravity

5. Jacobson’s “thermal origin” of gravity

MERA & Holography:

• Multi-scale Entanglement Renormalization Ansatz: approximateand efficient approach to build ground-state wavefunction ofcritical lattice models (Vidal; . . . )

• close connection between MERA and AdS/CFT (Swingle)

disentanglers :remove short-range

entanglement

isometries :decimate lattice

MERA & Holography:



• short-hand:

• question: where is the entanglement entropy?

UV

IR● ● ● ●

● ● ● ●

• introduce conjugate circuit• trace over exterior bonds

● ● ● ●

• SEE and SRen are unchanged by• 2k (=Leff) bonds → 1 bond in k layers• each layer contributes c/3 to SEE

SEE = c/3 log(Leff)

● ● ● ●

• SEE and SRen are unchanged by

B

● ● ● ●

• SEE and SRen are unchanged by

B

● ● ● ●

• only narrow band of circuitcontributes to SEE and SRen

MERA & Holography:



• question: where is the entanglement entropy?

only narrow band of circuit contributes to SEE and SRen

• common theme: long-range entanglement short-range entanglement

only narrow band of circuit contributes to SEE and SRen

• RT prescription: SEE depends on geometry near extremal surface

• MERA might provide insight for questions:

nature of extremization procedure?information in non-extremal surfaces? calculating Renyi entropies?

Conclusions:

• holographic entanglement entropy is part of an interestingdialogue has opened between string theorists and physicistsin a variety of fields (eg, condensed matter, nuclear physics, . . .)

• potential to learn lessons about issues in boundary theory

Dialogue:

Lots to explore!

• potential to learn lessons about issues in boundary theory

• potential to learn lessons about issues in quantum gravity in bulk

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