Review of Holographic (and other) computations of Entanglement Entropy, and its role in gravity
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1 Review of Holographic (and other) computations of Entanglement Entropy, and its role in gravity
2 Entanglement Entropy what is entanglement entropy? general tool; divide quantum system into two parts and use entropy as measure of correlations between subsystems in QFT, typically introduce a (smooth) boundary or entangling surface which divides the space into two separate regions integrate out degrees of freedom in outside region remaining dof are described by a density matrix calculate von Neumann entropy: A B
3 Entanglement Entropy remaining dof are described by a density matrix calculate von Neumann entropy: A R B result is UV divergent! dominated by short-distance correlations must regulate calculation: = short-distance cut-off = spacetime dimension careful analysis reveals geometric structure, eg,
4 Entanglement Entropy remaining dof are described by a density matrix calculate von Neumann entropy: A R B must regulate calculation: = short-distance cut-off = spacetime dimension leading coefficients sensitive to details of regulator, eg, find universal information characterizing underlying QFT in subleading terms, eg,
5 General comments on Entanglement Entropy: nonlocal quantity which is (at best) very difficult to measure no accepted experimental procedure in condensed matter theory: diagnostic to characterize quantum critical points, quantum spin fluids or topological phases in quantum information theory: useful measure of quantum entanglement (a computational resource)
6 General comments on Entanglement Entropy: Where did Entanglement Entropy come from? Sorkin 84: looking for origin of black hole entropy black holes recall that leading term obeys area law : suggestive of Bekenstein-Hawking formula if (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)
7 AdS/CFT Correspondence: Bulk: gravity with negative Λ in d+1 dimensions anti-de Sitter space holography Boundary: quantum field theory in d dimensions conformal field theory radius energy
8 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) AdS boundary A B boundary conformal field theory AdS bulk spacetime gravitational potential/redshift UV divergence because area integral extends to
9 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) AdS boundary A B cut-off in boundary CFT: cut-off surface UV divergence because area integral extends to finite result by stopping radial integral at large radius: short-distance cut-off in boundary theory:
10 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) AdS boundary A B cut-off in boundary CFT: cut-off surface central charge (counts dof) Area Law
11 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) AdS boundary A B cut-off in boundary CFT: cut-off surface general expression (as desired): (d even) universal contributions (d odd)
12 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 1) leading contribution yields area law 2) recover known results of Calabrese & Cardy for d=2 CFT (also result for thermal ensemble)
13 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 3) in a pure state and both yield same bulk surface V AdS boundary
14 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 3) in a pure state and both yield same bulk surface V cf: thermal ensemble pure state horizon in bulk AdS boundary AdS/CFT Dictionary: thermal bath black hole
15 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 4) Entropy of eternal black hole = entanglement entropy of boundary CFT & thermofield double (Headrick) thermofield double boundary CFT extremal surface = bifurcation surface
16 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 5) sub-additivity: (Headrick & Takayanagi) [ all other inequalities: Hayden, Headrick & Maloney]
17 Holographic Entanglement Entropy: (Ryu & Takayanagi `06) conjecture Extensive consistency tests: 4) Entropy of eternal black hole = entanglement entropy of boundary CFT & thermofield double 5) sub-additivity: (Headrick & Takayanagi) (Hayden, Headrick & Maloney) for more general holographic framework, expect includes corrections corrections
18 for more general holographic framework, expect some progress with classical higher curvature gravity: note is not unique! and is wrong choice! correct choice understood for Lovelock theories test with universal term for d=4 CFT: (deboer, Kulaxizi & Parnachev) (Hung, Myers & Smolkin) (Solodukhin) c a thermal entropy universal contribution
19 Topics currently trending in Holographic S EE : suggest C/F-theorems (Myers & Sinha; Liu & Mezei;...) diagnostic of holo-fermi surfaces (Ogawa, Takayanagi & Ugajin; Huijse, Sachdev & Swingle;... ) probe of holo-quantum quenches probe of holo-rg flows (Abajo-Arrastia, Aparicio, Lopez; Balasubramanian et al;...) (Albash & Johnson; Myers & Singh;...) probe of holo-insulator/superconductor transition probe of holo-fractionalized states probe of holo-qft in curved/desitter space (Albash & Johnson; Cai, He, Li & Li...) (Albash, Johnson & Macdonald; Hartnoll & Huijse,...) (Maldacena & Pimental) probe of holo-logarithmic CFT s (Grumiller, Rieldler, Rosseel & Zojer)
20 Lessons for bulk gravity theory:????
21 AdS/CFT Dictionary: Boundary: thermal plasma Bulk: black hole Temperature Energy Entropy Temperature Energy Entropy
22 Lessons for bulk gravity theory: (entanglement entropy) boundary = (entropy associated with extremal surface) bulk What Entropy?/Entropy of what? not black hole! not horizon! not causal domain! (eg, Hubney, Rangamani & Takayanagi) extremal surface What are the rules?
23 Proposal: Geometric Entropy (Bianchi & Myers) in a theory of quantum gravity, for any sufficiently large region with a smooth boundary in a smooth background (eg, flat space), leading contribution to entanglement entropy takes the form: A B in QG, (short range) quantum entanglement corresponding to area law is a signature of macroscopic spacetime geometry
24 Lessons for bulk gravity theory: (entanglement entropy) boundary = (entropy associated with extremal surface) bulk R&T construction assigns entropy to bulk regions with unconventional boundaries: not black hole! not horizon! not causal domain! indicates S BH applies more broadly! what about extremization? needed to make match above other surfaces might be expected to give other entropic measures of entanglement in boundary theory (Hubeny & Rangamani; Balasubramanian, McDermott & van Raamsdonk) extremal surface
25 Proposal: Geometric Entropy (Bianchi & Myers) in a theory of quantum gravity, for any sufficiently large region with a smooth boundary in a smooth background (eg, flat space), there is an entanglement entropy which takes the form: 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
26 Where did Entanglement Entropy come from? black holes Sorkin 84: looking for origin of black hole entropy recall that leading term obeys area law : suggestive of BH formula if (Sorkin `84; Bombelli, Koul, Lee & Sorkin; Srednicki; Frolov & Novikov) 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 dependent but for a given regulator should match!
27 BH Entropy ~ Entanglement Entropy (Susskind & Uglum) a beautiful killed by ugly facts?? seems matching is not always working?? both coefficients are regulator dependent but for a given regulator should match! (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)
28 BH Entropy ~ Entanglement Entropy (Susskind & Uglum) a beautiful killed by ugly facts?? result unsatisfying : both coefficients are regulator dependent but for a given regulator should match! where did bare term come from? formally off-shell method is precisely calculation of S EE (Susskind & Uglum; Callan & Wilczek; Myers & Sinha: extends to S wald ) challenge: understand microscopic d.o.f. of quantum gravity AdS/CFT: eternal black hole (or any Killing horizon) (Maldacena; van Raamsdonk et al; Casini, Huerta & Myers)
29 return to considerations of S EE of general region in some QFT first step in calculation of S EE is to determine A B encodes standard correlators, eg, if global vacuum: by causality, describes physics throughout causal domain
30 Entanglement Hamiltonian: A B hermitian and positive semi-definite, hence = modular or entanglement Hamiltonian formally can consider evolution by unfortunately is nonlocal and flow is nonlocal/not geometric
31 Entanglement Hamiltonian: A B hermitian and positive semi-definite, hence = modular or entanglement Hamiltonian explicitly known in only a few instances most famous example: Rindler wedge boost generator B A
32 Entanglement Hamiltonian: Can this formalism provide insight for area law term in S EE? A zoom in on infinitesimal patch of region looks like flat space looks like Rindler horizon assume Hadamard-like state correlators have standard UV sing s UV part of same as in flat space must have Rindler term:
33 Entanglement Hamiltonian: Can this formalism provide insight for area law term in S EE? for each infinitesimal patch: UV part of must be same as in flat space A must have Rindler term: Rindler yields area law; hence hence S EE must contain divergent area law contribution! invoke Cooperman & Luty: area law divergence matches precisely renormalization of : for any large region of smooth geometry!!
34 Entanglement Hamiltonian: Can this formalism provide insight for area law term in S EE? A Cooperman & Luty: area law divergence matches precisely renormalization of : for any large region of smooth geometry!! we can formulate a formal geometric argument analogous to the off-shell method for black holes (or Rindler horizons) challenge: understand microscopic d.o.f. of quantum gravity
35 Proposal: Geometric Entropy (Bianchi & Myers) in a theory of quantum gravity, for any sufficiently large region with a smooth boundary in a smooth background (eg, flat space), there is an entanglement entropy which takes the form: 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
36 MERA & Holography: Multi-scale Entanglement Renormalization Ansatz: approximate and efficient approach to build ground-state wavefunction of critical lattice models (Vidal;... ) close connection between MERA and AdS/CFT (Swingle) disentanglers: remove short-range entanglement isometries: decimate lattice
37 MERA & Holography: Multi-scale Entanglement Renormalization Ansatz: approximate and efficient approach to build ground-state wavefunction of critical lattice models (Vidal;... ) close connection between MERA and AdS/CFT (Swingle) short-hand: question: where is the entanglement entropy?
38 UV IR
39 introduce conjugate circuit trace over exterior bonds
40 introduce conjugate circuit trace over exterior bonds
41 S EE and S Ren are unchanged by 2 k (=L eff ) bonds 1 bond in k layers each layer contributes c/3 to S EE S EE = c/3 log(l eff )
42 S EE and S Ren are unchanged by
43 B S EE and S Ren are unchanged by
44 B only narrow band of circuit contributes to S EE and S Ren
45 B only narrow band of circuit contributes to S EE and S Ren
46 MERA & Holography: Multi-scale Entanglement Renormalization Ansatz: approximate and efficient approach to build ground-state wavefunction of critical lattice models (Vidal;... ) close connection between MERA and AdS/CFT (Swingle) question: where is the entanglement entropy? only narrow band of circuit contributes to S EE and S Ren RT prescription: S EE depends on geometry near extremal surface common theme: long-range entanglement short-range entanglement MERA might provide insight for questions: nature of extremization procedure? information in non-extremal surfaces? calculating Renyi entropies?
47 Conclusions: holographic entanglement entropy is part of an interesting dialogue has opened between string theorists and physicists in a variety of fields (eg, condensed matter, nuclear physics,...) Dialogue: potential to learn lessons about issues in boundary theory potential to learn lessons about issues in quantum gravity in bulk Lots to explore!
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