R enyi Entropy and Spectral Geometry Alexander Patrushev in - PowerPoint PPT Presentation

R´ enyi Entropy and Spectral Geometry Alexander Patrushev in collaboration with Dmitri Fursaev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research Strings 2014 Alexander Patrushev (JINR) RE and SG Princeton 1 / 9

Entanglement and R´ enyi entropy Take a system H with ρ and divide it into two subsystem H A and H B separated by a boundary (not necessarily physical) 1 1 − n ln tr ρ n S n = (1) A ρ A with integrated out DOF’s of B We measure correlation between A and B , measured on the entangling surface Σ Gentle coarse-graining ⇒ Universality Information theory (e.g. n → ∞ randomness extractors) Alexander Patrushev (JINR) RE and SG Princeton 2 / 9

PHYSICS String Theorists STRING THEORY Figure: String theory traced away People working on the border are most vulnerable. Alexander Patrushev (JINR) RE and SG Princeton 3 / 9

A replica method is based on calculation of the traces of density matrix ρ n powers Tr R ˆ R . Hard! Direct calculations using regularized metric Maldacena,Lewkowicz; Fursaev,A.P.,Solodukhin ds 2 = f ( r , b ) dr 2 + r 2 d τ 2 + [ h ij + r n cos ( τ ) K ij + r n sin ( τ ) K ij ] dx i dx j (2) ds 2 = f ( r , n ) dr 2 + r 2 d τ 2 + [ h ij + r cos ( τ ) K ij + r sin ( τ ) K ij ] dx i dx j (3) give the Weyl tensor up to O(n-1) ⇒ not good for Renyi entropy Conformal transformation maps entanglement entropy on a flat space-time to thermal entropy on S 1 × H 3 Casini,Huerta,Myers (Good for Renyi entropy for a spherical entangling surface, possible corrections Rosenhaus,Smolkin) Numerical methods Holography? Alexander Patrushev (JINR) RE and SG Princeton 4 / 9

General expression for the logarithmic term in the R´ enyi entropy in ( 3 + 1 ) -dimensional CFT Fursaev 2 k 2 � − f c ( n ) Σ = f a ( n ) E 2 + f b ( n ) � Tr k 2 − 1 � � � S n W ab ab log ǫ , (4) 180 240 π 240 π Σ Σ Σ where f a , b , c ( n ) only depend on n Numerics f b ( q ) = f c ( q ) Lee,McGough,Safdi Alexander Patrushev (JINR) RE and SG Princeton 5 / 9

R´ enyi entropy for excited states Takayanagi et al � Tr ρ n = D φ exp ( − I n [ φ, J ]) (5) 1 S n ( J ) = 1 − n ( ln Z n − n ln Z 1 ) (6) Finite entropy 1 � ∆ S n = J n ( G n − ( n − 1 ) G 1 ) J n (7) 2 ( n − 1 ) M \ G n is a Green function for M n � G α = − dsK α ( s ) (8) Sommerfeld formula K α ( t ) = 1 � dz � cot π � x ( z ) , x ′ ( 0 ) | t α ( z − t ) K (9) 2 α B Generalization for squashed geometries. Alexander Patrushev (JINR) RE and SG Princeton 6 / 9

Generalization of the Sommerfeld formula for squashed geometries � dw , K α ( t ) = 1 cot π � � x ( z ) , x ′ ( 0 ) | t α ( w − ∆ τ ) F (10) 2 α B where F ( x , x ′ , ω | t ) is a new kernel F ( x , x ′ , ω | t ) = K ( x ( ω ) , x ′ ( ω ) | t ) e − s ( x , x ′ ,ω | t ) / ( 4 t ) ( 1 + A ( x , x ′ , ω | t )) (11) with a consistency condition F ( x , x ′ , ω | t ) | ω = τ − τ ′ = K ( x ( ω ) , x ′ ( ω ) | t ) | α = 2 π (12) Alexander Patrushev (JINR) RE and SG Princeton 7 / 9

Heat kernel coefficient a 4 gives a logarithmic term K β ( t ) ∼ t − 1 a 2 + a 4 + . . . ( t − dependent ) (13) a 4 → a ( γ n ) f a ( n ) + b ( γ n ) f b ( n ) + c ( γ n ) f c ( n ) , (14) γ n = 2 π α n Alexander Patrushev (JINR) RE and SG Princeton 8 / 9

Figure: Thank you for your attention. Your Questions, Please. Alexander Patrushev (JINR) RE and SG Princeton 9 / 9