Entanglement entropy: logarithmic terms Sergey Solodukhin Institut Denis Poisson (Tours) Talk at The Vacuum of the Universe IV, 10-13 June 2019 Sergey Solodukhin Entanglement entropy: logarithmic terms
Outline of the talk Introduction Introduction: history, definition and methods of computation Entanglement entropy of black holes Puzzle of non-minimal coupling Holography: entropy as minimal surface Log terms in EE of a Conformal Field Theory (CFT) Why log terms might be interesting? Conclusions Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: 1976: Werner Israel Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: 1976: Werner Israel 1985: ’t Hooft (brick wall model) Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: 1976: Werner Israel 1985: ’t Hooft (brick wall model) 1986: Bombelli, Koul, Lee and Sorkin Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: 1976: Werner Israel 1985: ’t Hooft (brick wall model) 1986: Bombelli, Koul, Lee and Sorkin First big wave, 1993 - 1998: Players in that period include (in a chaotic order) Srednicki , Callan, Wilzcek, Susskind, Uglum, Jacobson, Larsen, Holzhey, Frolov, Novikov, Barvinsky, Zelnikov, Brustein, Mann, Dowker, Emparan, Kabat, Strassler, Borbon, Fursaev, Zerbini, Vanzo, Kirsten, Myers, Demers, Lafrance, Dabholkar, SS . . . Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some history Some major periods in study of entanglement entropy: Earlier works: 1976: Werner Israel 1985: ’t Hooft (brick wall model) 1986: Bombelli, Koul, Lee and Sorkin First big wave, 1993 - 1998: Players in that period include (in a chaotic order) Srednicki , Callan, Wilzcek, Susskind, Uglum, Jacobson, Larsen, Holzhey, Frolov, Novikov, Barvinsky, Zelnikov, Brustein, Mann, Dowker, Emparan, Kabat, Strassler, Borbon, Fursaev, Zerbini, Vanzo, Kirsten, Myers, Demers, Lafrance, Dabholkar, SS . . . Second big wave, 2006 - present: Ryu and Takayanagi, and many people after . . . Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: definition | ψ > = � i , a ψ ia | A > i | B > a and density matrix A pure (vacuum) state ρ 0 ( A , B ) = | ψ >< ψ | | A > states are inside surface Σ and | B > are outside of Σ Density matrix ρ B = Tr A ρ 0 ( A , B ) and entropy S B = − Tr ρ B ln ρ B Since Tr ρ k A = Tr ρ k B entropy S A = S B depends on geometry of separation surface Σ and space-time geometry near Σ That is why in earlier years it was called geometric entropy Bombelli, Koul, Lee and Sorkin ’86; Srednicki ’93; Frolov and Novikov ’93 Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: some properties In Quantum Field Theory entanglement entropy is UV divergent (function of UV cut-off ǫ ) to leading order EE is proportional to area of Σ S ∼ A (Σ) S ∼ c if d > 2 and 6 ln (1 /ǫ ) if d = 2 ǫ d − 2 due to short-distance correlations across Σ Bombelli, Koul, Lee and Sorkin ’86; Srednicki ’93; Holzhey, Larsen and Wilzcek ’94 c In 2d CFT c is central charge < T > = 48 π R Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: general structure of EE More generally, in d-dimensional curved space-time (with no boundary) EE is a Laurent series S = s d − 2 ǫ d − 2 + s d − 4 ǫ d − 4 + · · · + s d − 2 n ǫ d − 2 n + · · · + s 0 ln ǫ + s ( g ) ˆ � R l k 2 p s d − 2 − 2 n = Σ ( l + p )= n R is Riemann curvature and k is extrinsic curvature of Σ Since there are 2 normal vectors to Σ only even powers of k may appear Logarithmic term s 0 is non-zero if d is even If space-time has boundary ∂ M and if Σ intersects ∂ M then the story is different: Log term may appear in any dimension d (odd or even) Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: replica method In Quantum Field Theory and in presence of rotational symmetry (in sub-space orthogonal to Σ) Tr ρ n = Z [ C n ] is partition function on conical space with angle deficit 2 π (1 − n ) at surface Σ so that EE is computed by differentiating w.r.t. n of effective action W ( n ) = − ln Z ( n ) on conical space S = ( n ∂ n − 1) W ( n ) | n =1 Heat kernel method (field operator D = −∇ 2 + ξ R ) ˆ ∞ W = − 1 ds 1 � ( a reg + a Σ k ) s k s Tr K ( s ) , Tr K M n = k (4 π s ) d / 2 2 ǫ 2 k =0 1 − α 2 1 = π ˆ a Σ 1 3 α Σ 1 − α 2 1 − α 4 2 = π ( 1 π ˆ ˆ a Σ 6 − ξ ) R − ( R aa − 2 R abab ) α 3 3 α 180 Σ Σ McKean and Singer ’67; Cheeger ’83; Dowker ’77; Fursaev ’94 Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: distributional geometry of squashed cones If no rotational symmetry (extrinsic curvature k of Σ is non-zero) one considers squashed cones ˆ ˆ ˆ R + 4 π (1 − n ) R = n 1 Σ M n M D.D. Sokolov and A. Starobinsky ’77 ˆ ˆ ˆ R 2 = n R 2 + 8 π (1 − n ) R M n M Σ ( R aa − 1 ˆ ˆ ˆ R 2 R 2 2 k 2 ) µν = n µν + 4 π (1 − n ) M n M Σ ˆ ˆ ˆ R 2 R 2 ( R abab − Tr k 2 ) αβµν = n αβµν + 8 π (1 − n ) M n M Σ R ab = R µν n µ a n ν R abab = R αβµν n α a n β b n µ a n ν and a b Fursaev and SS ’94; Fursaev, Patrushev and SS ’13 Sergey Solodukhin Entanglement entropy: logarithmic terms
Introduction: distributional geometry of squashed cones Topological Euler number χ 4 [ M n ] = n χ 4 [ M ] + (1 − n ) χ 2 [Σ] Conformal invariant ˆ ˆ ˆ W 2 = n W 2 + 8 π (1 − n ) [ W abab − Tr ˆ k 2 ] M n M Σ 2 γ µν Tr k a , a = 1 , 2 is conformal invariant constructed from extrinsic ˆ µν − 1 k a µν = k a curvature. Fursaev and SS ’94; Fursaev, Patrushev and SS ’13 Sergey Solodukhin Entanglement entropy: logarithmic terms
Entanglement entropy of black holes Historically the study of EE was motivated by attempts to find a stat. mechanical explanation of Bekenstein-Hawking entropy If Σ is black hole horizon then its extrinsic curvature vanishes k a = 0 , a = 1 , 2 rotational symmetry is generated by Killing vector S d =4 = A (Σ) 1 ˆ [ R (1 − 6 ξ ) − 1 48 πǫ 2 − 5 ( R aa − 2 R abab )] ln ǫ 144 π Σ Myers, Demers, Lafrance ’94; SS ’94 EE of the Schwarzschild black hole S Sch = A (Σ) 48 πǫ 2 + 1 45 ln r + ǫ for any value of ξ SS ’94 This is entire entanglement entropy including UV finite part! Sergey Solodukhin Entanglement entropy: logarithmic terms
Puzzle of non-minimal coupling If Riemann curvature appears in field operator (as in D = −∇ 2 + ξ R ) should we take into account its distributional part when consider on conical space M n ? If we do then (for scalar field) one finds for heat kernel ˆ a reg a Σ k → a Σ k − 4 πξ (1 − n ) k − 1 Σ and for entropy (SS ’95) S con = A (Σ) 1 [ R (1 − 6 ξ ) 2 − 1 ˆ 48 πǫ 2 (1 − 6 ξ ) − 5 ( R aa − 2 R abab )] ln ǫ 144 π Σ In Log term no changes if ξ = 1 / 6 (conformal case) since a reg = 0 in this case 1 s 0 is invariant under conformal rescaling preserving horizon No modification in Log term for the Schwarzschild black hole Area term is not positive definite in general. That means this entropy does not correspond to a well-defined density matrix. SS ’95; Larsen, Wilzcek ’95 Sergey Solodukhin Entanglement entropy: logarithmic terms
Puzzle of non-minimal coupling: Kabat’s contact terms Similar story for gauge fields (contact terms of D. Kabat ’95) S con = A [Σ] 8 πǫ 2 ( d − 2 − 1) 6 is negative in dimensions d < 8 Sergey Solodukhin Entanglement entropy: logarithmic terms
Holography: entropy as minimal surface Holographic Entanglement Entropy t g g is minimal surface Area ( ) B S that bounds = S F 4 G N A N B g x r Ryu-Takayanagi (06) 1 Sergey Solodukhin Entanglement entropy: logarithmic terms
Holography: entropy as minimal surface UV/IR duality t c 1 = S ln( ), e 6 B = F d 2 e S Area ( ) , A S ! e - d 2 N > d 2 e B g x r Ryu-Takayanagi (06) 1 Sergey Solodukhin Entanglement entropy: logarithmic terms
Holography: entropy as minimal surface Graham-Witten surface anomaly G Graham-Witten (99) S r S Area ( ) G = + S e + Area ( ) c ( )ln .. e 2 Graham-Witten anomaly 1 Sergey Solodukhin Entanglement entropy: logarithmic terms
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