review of holographic (and other) computations of...
TRANSCRIPT
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:
Entanglement Entropy
AB
• remaining dof are described by a density matrix
calculate von Neumann entropy:
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,
Entanglement Entropy
AB
• remaining dof are described by a density matrix
calculate von Neumann entropy:
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
Holographic Entanglement Entropy:
AB
AdS boundary
(Ryu & Takayanagi `06)
cut-off in boundary CFT:
central charge(counts dof) “Area Law”
cut-off surface
Holographic Entanglement Entropy:
AB
AdS boundary
(Ryu & Takayanagi `06)
cut-off in boundary CFT:
cut-off surface
general expression (as desired):
(d even)
(d odd)universal contributions
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
1) leading contribution yields “area law”
2) recover known results of Calabrese & Cardyfor d=2 CFT
(also result for thermal ensemble)
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
3) in a pure state
and both yield same bulk surface V
AdSboundary
and both yield same bulk surface V
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
3) in a pure state
and both yield same bulk surface V
AdSboundary
and both yield same bulk surface V
cf: thermal ensemble ≠ pure state horizon in bulk
AdS/CFT Dictionary: black holethermal bath
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
4) Entropy of eternal black hole =entanglement entropy of boundary CFT & thermofield double
(Headrick)(Headrick)
boundaryCFT
thermofielddouble
extremal surface =bifurcation surface
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
5) sub-additivity:(Headrick & Takayanagi)
[ “all” other inequalities: Hayden, Headrick & Maloney]
Holographic Entanglement Entropy:
conjecture
Extensive consistency tests:
(Ryu & Takayanagi `06)
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
Lessons for bulk gravity theory:
(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:
Lessons for bulk gravity theory:
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)
• evidence comes from several directions:
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)
both coefficients are regulator dependentbut for a given regulator should match!
• 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)
both coefficients are regulator dependentbut for a given regulator should match!
• “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
Entanglement Hamiltonian:Can this formalism provide
insight for “area law” term in S EE?
• 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
Entanglement Hamiltonian:Can this formalism provide
insight for “area law” term in S EE?
• 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:
• evidence comes from several directions:
(Bianchi & Myers)
• evidence comes from several directions:
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:
• 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)
• short-hand:
• question: where is the entanglement entropy?
UV
IR● ● ● ●
● ● ● ●
• introduce conjugate circuit• trace over exterior bonds
● ● ● ●
• 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
B
● ● ● ●
• only narrow band of circuitcontributes to SEE and SRen
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)
• 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