Entanglement entropy in two-dimensional conformal field theory - PowerPoint PPT Presentation

Entanglement entropy in two-dimensional conformal field theory Paweł Grabiński University of Wrocław 22 July 2016 Entanglement in 2dCFT Paweł Grabiński 1/12

Entanglement ◮ Density operator: ρ = | ψ � ⊗ � ψ | ˆ ◮ Reduced density operator: ρ A = tr A ˆ ˆ ρ ◮ Separable subsystems: | ψ � AB = | φ � A ⊗ | χ � B ◮ Entangled subsystems: √ p i | i � A ⊗ | i � B � | ψ � AB = i ◮ Entropy of entanglement(von Neuman Entropy): � S A = − tr (ˆ ρ A log ˆ ρ A ) = − p i log p i i Entanglement in 2dCFT Paweł Grabiński 2/12

Density operator in QFT Density operator in Quantum Field Theory can be defined as: � ρ ( φ 1 , φ 2 ) = Z − 1 D φ e − S E [ φ ] (1) F ( φ 1 ,φ 2 ) φ ( β, x ) τ [ 0 , β ] × R S 1 × R x x β φ ( 0 , x ) = φ ( β, x ) φ ( 0 , x ) τ 0 Entanglement in 2dCFT Paweł Grabiński 3/12

Reduced density operator in QFT Reduced density operator can be defined as: � ρ ( φ 1 , φ 2 ) = Z − 1 D φ e − S E [ φ ] S\ A φ (0 , x ) = φ ( β, x ) φ ( β, x ) � = φ (0 , x ) . B A . B τ x Entanglement in 2dCFT Paweł Grabiński 4/12

2d Conformal Field Theory ◮ Conformal Transformation: µν ( x ′ ) = ∂ x α ∂ x β g ′ ′ ν g µν ( x ) = Ω( x ) g µν ( x ) , µ, ν = 1 , 2 ′ µ ∂ x ∂ x ◮ Primary Fields Transformation in CFT( w = f ( z ) ): � − ¯ � − h � d ¯ h � dw w φ ′ ( w , ¯ w ) = φ ( z , ¯ z ) d ¯ dz z ◮ Global Ward’s Identities ( i = − 1 , 0 , 1 ) from SL ( 2 , C ) : n � � � z i z j ∂ j + h j ( i + 1 ) � φ ( z 1 , ¯ z 1 ) . . . φ ( z j , ¯ z j ) . . . φ ( z n , ¯ z n ) � = 0 j j = 1 ◮ 2-point correlation function from GWI: z 1 ) � = A ( z 1 − z 2 ) − 2 h ( ¯ z 2 ) − 2 ¯ h � φ ( z 1 , ¯ z 1 ) φ ( z 1 , ¯ z 1 − ¯ Entanglement in 2dCFT Paweł Grabiński 5/12

Entropy and Replica trick ◮ Renyi Entropy: 1 S ( n ) ρ n = − n − 1 log tr ˆ A A ◮ Entanglement Entropy: = − d n → 1 S ( n ) ρ n S A = lim dn tr ˆ A | n = 1 a ◮ Replica trick ( T = 0 case): Entanglement in 2dCFT Paweł Grabiński 6/12

Entanglement entropy at T = 0 ◮ From non-trivial geometry to Twist fields: n � T ( w , i ) � L , R n = n � T ( w ) � L , R n = � T n ( z ) T ( a , a ) ¯ T ( b , b ) � L n , C � �T ( a , a ) ¯ T ( b , b ) � L n , C i = 1 ◮ Our conformal transformation: � 1 � w − a n f : R n → C , C ∋ z = f ( w ) = w ∈ R n , w − b ◮ From transformation rule of energy-momentum tensor: � dz � 2 T ( z ) + c T ( w ) = 12 { z , w } dw � T ( z ) � L n , C = 0 ⇒ � T ( w ) � L , R n = c ( n 2 − 1 ) ( a − b ) 2 24 n 2 ( w − a ) 2 ( w − b ) 2 Entanglement in 2dCFT Paweł Grabiński 7/12

Entropy for interval A = [ a , b ] at T = 0 ◮ Local Ward’s Identities: � T ( z ) φ ( z 1 , ¯ z 1 ) . . . φ ( z n , ¯ z n ) � = n � 1 � h i � ( z − z i ) 2 + ∂ i � φ ( z 1 , ¯ z 1 ) . . . φ ( z n , ¯ z n ) � z − z i i = 1 h = c ( n 2 − 1 ) ◮ From Local Ward’s Identities we get: h = ¯ 24 n ◮ Trace of the reduced density operator takes form( α ( n ) = A Z n ): A } = �T ( a , a ) ¯ T ( b , b ) � L n , C = α ( n )( a − b ) − c ( n 2 − 1 ) tr { ρ n 6 n Z n ◮ Entropy of interval A = [ a , b ] at T = 0: S A = c α ( 1 ) ln ( a − b ) − d α dn | n = 1 , α ( 1 ) = 1 3 Entanglement in 2dCFT Paweł Grabiński 8/12

Results for interval in T > 0 ◮ Field transformation rule: 2 h 2 h � � � � dw dw �T ( u , u ) ¯ �T ( a , a ) ¯ � � � � T ( v , v ) � L , cyl = T ( b , b ) � L n , C � � � � dz dz � � � � a b ◮ Mapping g : C → R × S z = g ( w ) = 2 π β log ( w ) ◮ Form of correlation function: �� − 4 h � β � π �T ( u , u ) ¯ T ( v , v ) � L , cyl = A β ( u − v ) π sh ◮ Entropy of interval A = [ a , b ] in T > 0: � β � π �� S A = c − d α 3 ln β ( u − v ) dn | n = 1 π sh Entanglement in 2dCFT Paweł Grabiński 9/12

Interval on a circle at T = 0 Performing rotation of cylinder( β = iL ): S A = c � L � π �� − d α 3 ln π sin L ( u − v ) dn | n = 1 Entropy of entanglment 0.2 0.1 l 0.2 0.4 0.6 0.8 1.0 L � 0.1 � 0.2 � 0.3 Entanglement in 2dCFT Paweł Grabiński 10/12

Statistical models correspondance 2-dimensional Conformal Field Theory � Statistical Models at Critical Point 0.8 0.6 S n (L)-log(L/ π)/3 0.4 n=1 0.2 n=2 0 n=3 n=4 -0.2 n=5 n=6 -0.4 0 0.2 0.4 0.6 0.8 1 n/L Example: Quantum 1d Ising model with transverse field with hamiltonian: N − 1 � J S x i S x i + 1 + hS z � � H = − i i = 1 Entanglement in 2dCFT Paweł Grabiński 11/12

Literature Thank You! P. Di Francesco, P. Mathieu, D. Senechai, Conformal Field Theory , Springer-Verlag, New York 1997 P. Calabrese, J. Cardy, Entanglement Entropy and Quantum Field Theory F. Leja, Analytic and harmonic functions , PTW, Warsaw 1952 Entanglement in 2dCFT Paweł Grabiński 12/12