Entanglement entropy: Entanglement entropy hints from the two - PowerPoint PPT Presentation

Plan of the talk Entanglement entropy: Entanglement entropy hints from the two intervals case Erik Tonni MIT based on P. Calabrese, J. Cardy and E.T.; [0905.2069] (JSTAT) P. Calabrese, J. Cardy and E.T.; [1011.5482] E.T.; [1011.0166] GGI, Firenze, October 2010

Plan of the talk Introduction and definition of the entanglement entropy Replica trick Twist fields Entanglement entropy of one interval in CFT Entanglement entropy of two disjoint intervals for the free compactified boson (Luttinger liquid) Special cases and special regimes Analytic continuation Comparison with numerical data from XXZ spin chain Ising model Holographic entanglement entropy Conclusions and open problems

Entanglement entropy: definition Quantum system ( H ) in the ground state | Ψ � Tr ρ n = 1 Density matrix ρ = | Ψ �� Ψ | = ⇒ Two observers: each one measures only a subset of a H = H A ⊗ H B complete set of cummuting observables B B A A B A B ρ A = Tr B ρ ρ A = Tr T B ρ A ’s reduced density matrix Entanglement entropy ≡ Von Neumann entropy of ρ A S A = − Tr A ( ρ A log ρ A ) It measures the amount of information shared by A and B

Replica trick ∂ [Holzhey, Larsen, Wilczek, NPB (1994)] ∂n Tr ρ n T ρ n S A = − lim n Tr A A n → 1 [Calabrese, Cardy, JSTAT (2004)] QFT with Hamiltonian H . Density matrix ρ in a thermal state at temperature T = 1 /β φ x τ = β � � � ρ ( { φ x }|{ φ ′ x ′ } ) = Z − 1 δ ( φ ( y, 0) − φ ′ δ ( φ ( y, β ) − φ x ) e − S E [ dφ ( y, τ )] x ′ ) τ = 0 x x ′ φ ′ x ′ Z = Tr e − βH . The trace sews together the edges at τ = 0 and τ = β providing a cylinder with circumference of length β . ρ A = Tr B ρ The trace Tr B sews together only the points / ∈ A . . . . . . . u 1 v 1 u N v N Open cuts are left along the β disjoint intervals ( u j , v j ). A = ( u 1 , v 1 ) ∪ · · · ∪ ( u N , v N )

Replica trick and Riemann surfaces n copies of the cylinder above sewed together cyclically along the cuts ∂ Z n ( A ) ” ρ ij A ρ jk A ρ kl A ρ li A ” = S A = − lim Z n ∂n n → 1 Tr ρ n A as a partition function on the n sheeted Riemann surface R n,N � � � � Z R n,N = [ dϕ 1 · · · dϕ n ] C exp − dzd ¯ z ( L [ ϕ 1 ]( z, ¯ z ) + . . . + L [ ϕ n ]( z, ¯ z )) C uj ,vj C ϕ i ( x, 0 + ) = ϕ i +1 ( x, 0 − ) C u j ,v j : R 3 , 1 N � x ∈ [ u j , v j ] i = 1 , . . . , n j =1 [Cardy, Castro-Alvaredo, Doyon, JSP (2007)]

Twist fields Global symmetry σ : i �→ i + 1 mod n � � dxdy L [ σϕ ]( x, y ) = dxdy L [ ϕ ]( x, y ) σ − 1 : i + 1 �→ i mod n T n ≡ T σ The twist fields implement this global symmetry ˜ T n ≡ T σ − 1 Z R n,N = �T n ( u 1 , 0) ˜ T n ( v 1 , 0) · · · T n ( u N , 0) ˜ T n ( v N , 0) � L ( n ) , C n − 1 n − 1 � ˜ � ˜ T n = T n,k T n = T n,k k =0 k =0 n − 1 � �T n,k ( u 1 , 0) ˜ T n,k ( v 1 , 0) · · · T n,k ( u N , 0) ˜ Z R n,N = T n,k ( v N , 0) � L ( n ) , C k =0

Boundary conditions and twist fields R 3 , 1 Boundary conditions: ϕ j ( e 2 πi z, e − 2 πi ¯ z ) = ϕ j − 1 ( z, ¯ z ) Linear combinations of basic fields which diagonalize the twist n e 2 πi k � n j ϕ j ϕ k ≡ ˜ k = 0 , 1 , . . . , n − 1 j = 1 z ) = e 2 πi k θ k ≡ e 2 πi k ϕ k ( e 2 πi z, e − 2 πi ¯ n ˜ ˜ ϕ k ( z, ¯ z ) = θ k ˜ ϕ k ( z, ¯ z ) n Branch-point twist field T n,k in the origin [Dixon, Friedan, Martinec, Shenker, NPB (1987)] [Zamolodchikov, NPB (1987)]

Entanglement of a single interval Two-point function of twist fields for a free complex boson ϕ [Dixon, Friedan, Martinec, Shenker, NPB (1987)] 1 � � n = 1 k 1 − k �T k,n ( u ) ˜ n = ¯ T k,n ( v ) � ∝ ∆ k ∆ k | u − v | 4∆ k/n 2 n n Partition function on R n, 1 n − 1 n − 1 c n � � �T k,n ( u ) ˜ Z R n, 1 = Z k,n = T k,n ( v ) � = 3 ( n − 1 1 n ) | u − v | k =0 k =0 Entanglement entropy of a single interval for the free real boson c = 1 n =1 = 1 n =1 = 3 log � = 1 lo 3 log � S A = − ∂ n Tr ρ n T ρ n � � � a + c ′ a + c ′ S A = − ∂ n ∂ Tr A A � � 1 1 [Holzhey, Larsen, Wilczek, NPB (1994)]

Entanglement of two disjoint intervals [Calabrese, Cardy and E.T.; JSTAT (2009)] = ⇒ R n, 2 A = A 1 ∪ A 2 = [ u 1 , v 1 ] ∪ [ u 2 , v 2 ] e.g.: R 3 , 2 Four-point function of twist fields for a free, real, compactified boson ϕ � c 6 ( n − 1 /n ) ) � | u 1 − u 2 || v 1 − v 2 | Tr ρ n A ≡ Z R n, 2 = c 2 F ( x ) F n F n ( x ) n | u 1 − v 1 || u 2 − v 2 || u 1 − v 2 || u 2 − v 1 | � �� � Z W x = ( u 1 − v 1 )( u 2 − v 2 ) R n, 2 ( u 1 − u 2 )( v 1 − v 2 ) [Calabrese, Cardy, JSTAT (2004)]

Computation of F n ( x ) (I) Compactification condition ϕ j ( e 2 πi z, e − 2 πi ¯ m j ∈ Z + i Z z ) = ϕ j − 1 ( z, ¯ z ) + R ( m j, 1 + im j, 2 ) n � θ j ϕ k ( e 2 πi z, e − 2 πi ¯ θ k ≡ e 2 πi k ˜ z ) = θ k ˜ ϕ k ( z, ¯ z ) + R k m j ξ ∈ R Λ k n n j = 1 Partition function on R n, 2 from the four-point function of twist fields n − 1 � � Z qu k,n Z cl Z R n, 2 = [Dixon, Friedan, Martinec, Shenker, NPB (1987)] k,n m ∈ Z 2 n k =0 � �� � � � n − 1 � � π k | ξ 1 | 2 β k/n + | ξ 2 | 2 − 2 gπ sin const n F n ( x ) = � 2 exp � n β k/n β k/n F k/n ( x ) m ∈ Z 2 n k = 0 β y ≡ F y (1 − x ) F y ( x ) ≡ 2 F 1 ( y, 1 − y ; 1; x ) F y ( x ) Z cl does not contribute in the decompactification limit

Computation of F n ( x ) (II) � � � �� 2 � � const m t · Ω · m + m t · � F n ( x ) = exp i π Ω · m � � 2 � n − 1 k = 0 β k/n F k/n ( x ) m ∈ Z n � � � � � � 1 � � n − 1 n − 1 � � Ω rs ≡ 2 gR 2 i π k 2 π k Ω rs ≡ 2 gR 2 i π k 2 π k � sin β k n cos n ( r − s ) sin cos n ( r − s ) n n n n β k k = 0 k = 0 n r, s = 1 , . . . , n Regularize the sum by eliminating the contribution of the eigenvalue generating the kernel of both Ω and � Ω (non trivial step!) Riemann-Siegel theta function � � � iπ m t · Γ · m + 2 πi m t · z Θ( z | Γ) ≡ exp z ∈ C G m ∈ Z G Γ is a symmetric, G × G matrix with positive imaginary part

Computation of F n ( x ) : main result � � � � � � � � n − 1 n − 1 � � Γ rs ≡ 2 i π k 2 π k Γ rs ≡ 2 i π k 1 2 π k � sin β k/n cos n ( r − s ) sin cos n ( r − s ) n n n n β k/n k = 1 k = 1 r, s = 1 , . . . , n − 1 � � � � � � � � η ≡ gR 2 0 | η � 0 | η � ] 2 ] 2 [Θ [Θ 0 | η Γ 0 | η Γ Θ Θ Γ Γ F ( x ) = const F n F n ( x ) = const � n − 1 � n − 1 k = 1 F k k = 1 F k/n ( x ) F k/n (1 − x ) F /n ( x ) F k F /n (1 − x ) nasty F n ( x ) is invariant under x ↔ 1 − x n dependence Fix the constant s.t. F n (0) = 1 Riemann-Siegel theta function manipulations Final result � � 2 � � � � Θ 0 | η Γ Θ 0 | Γ /η F n ( x ) = � � 2 Θ 0 | Γ F n ( x ) is invariant under η ↔ 1 /η

Special cases n = 2 [Furukawa, Pasquier, Shiraishi, PRL (2009)] [Zamolodchikov, NPB (1987)] � θ 3 ( τ 1 / 2 η ) θ 3 ( τ 1 / 2 /η ) � 2 F 2 ( x ) = τ 1 / 2 = iβ 1 / 2 θ 2 3 ( τ 1 / 2 ) � � n = 3 √ Γ = τ 1 / 3 2 − 1 √ γ = 3 τ 1 / 3 − 1 2 3 � � ηγ � 2 � ηγ � 2 � ηγ � 2 � 1 θ 2 ( ηγ ) 2 θ 2 + θ 3 ( ηγ ) 2 θ 3 + θ 4 ( ηγ ) 2 θ 4 F 3 ( x ) = 4[ F 1 / 3 ( x )] 4 3 3 3 � γ � γ � γ � � γ � 2 � 2 � γ � 2 � 2 � γ � 2 � 2 � × θ 2 θ 2 + θ 3 θ 3 + θ 4 θ 4 η 3 η η 3 η η 3 η . . .

Special regimes and a generalization decompactification regime: large η (recall the symmetry η ↔ 1 /η ) η n − 1 F n ( x ) = � n − 1 k =1 F k/n ( x ) F k/n (1 − x ) In this regime we can perform the analytic continuation n → 1. x → 0 regime n − 1 � 2( n − l ) F n ( x ) = 1 + x min( η, 1 /η ) �� 2min( η, 1 /η ) + . . . � � π l 2 n sin l = 1 n n − 1 � � � R 1 m ( p ) l, 1 + i R 2 m ( p ) θ l different compactification radii ξ p = k l, 2 l =0 � Θ(0 | η 1 Γ) Θ(0 | Γ /η 1 ) � � Θ(0 | η 2 Γ) Θ(0 | Γ /η 2 ) � F n ( x ) = Θ(0 | Γ) 2 Θ(0 | Γ) 2

Analytic continuation decompactification regime: large η (recall the symmetry η ↔ 1 /η ) η n − 1 F n ( x ) = � n − 1 k =1 F k/n ( x ) F k/n (1 − x ) Useful representation: C iL � n − 1 � dz D n ( x ) = log F k/n ( x ) = 2 πiπ cot( πz ) log F z/n ( x ) n C k =1 − iL � � i ∞ � 1 ( x ) ≡ − ∂D n ( x ) dz πz � D ′ = sin 2 πz log F z ( x ) � ∂n i − i ∞ n =1 Mutual information I A 1 : A 2 ≡ S A 1 + S A 2 − S A 1 ∪ A 2 2 ln η + D ′ 1 ( x ) + D ′ A 1 : A 2 � − 1 1 (1 − x ) I A 1 : A 2 ( η ≪ 1) − I W 2

Comparison with the numerical data Exact diagonalization of the XXZ spin chain in a magnetic field (up to L = 30) [Furukawa, Pasquier, Shiraishi, PRL (2009)] L � � � η = 1 − 1 ∆ ∈ ( − 1 , 1] } j +1 + S y j S y S x j S x j +1 + ∆ S z j S z j +1 − h S z π arccos ∆ H ≡ j h = 0 j =1 x = 1 / 2 η = 0 . 295 η = 0 . 369 η = 0 . 436 x = 1 / 4 η = 0 . 5 2 ln η + D ′ 2 ln η + D ′ 1 ( x ) + D ′ 1 ( x ) + D ′ A 1 : A 2 � − 1 1 : A 2 � − 1 1 (1 − x ) 1 (1 − x ) I A 1 : A 2 ( η ≪ 1) − I W 1 : A 2 ( η ≪ 1) − I W confirms the formula I I A I A 2 2