Entanglement entropy of disconnected regions in Conformal Field - PowerPoint PPT Presentation

Entanglement entropy of disconnected regions in Conformal Field Theories Pasquale Calabrese Dipartimento di Fisica Universit` a di Pisa GGI workshop on AdS/CFT, Florence, November 2010 Mainly joint work with John Cardy also V Alba, M Campostrini, F Essler, M. Fagotti, A Lefevre, J Moore, B Nienhuis, I Peschel, L Tagliacozzo, E Tonni Review: PC & JC JPA 42, 504005 (2009) Pasquale Calabrese Entanglement and CFT

Entanglement: what is it? Quantum system in a pure state | Ψ � The density matrix is ρ = | Ψ �� Ψ | ( Tr ρ n = 1) H = H A ⊗ H B Alice can measure only in A, while Bob in the remainder B Pasquale Calabrese Entanglement and CFT

Entanglement: what is it? Quantum system in a pure state | Ψ � The density matrix is ρ = | Ψ �� Ψ | ( Tr ρ n = 1) H = H A ⊗ H B Alice can measure only in A, while Bob in the remainder B Alice measures are entangled with Bob’s ones: Schmidt deco � � c 2 | Ψ � = c n | Ψ n � A | Ψ n � B c n ≥ 0 , n = 1 n n Pasquale Calabrese Entanglement and CFT

Entanglement: what is it? Quantum system in a pure state | Ψ � The density matrix is ρ = | Ψ �� Ψ | ( Tr ρ n = 1) H = H A ⊗ H B Alice can measure only in A, while Bob in the remainder B Alice measures are entangled with Bob’s ones: Schmidt deco � � c 2 | Ψ � = c n | Ψ n � A | Ψ n � B c n ≥ 0 , n = 1 n n If c 1 = 1 ⇒ | Ψ � unentangled If c i all equal ⇒ | Ψ � maximally entangled Pasquale Calabrese Entanglement and CFT

Entanglement: what is it? Quantum system in a pure state | Ψ � The density matrix is ρ = | Ψ �� Ψ | ( Tr ρ n = 1) H = H A ⊗ H B Alice can measure only in A, while Bob in the remainder B Alice measures are entangled with Bob’s ones: Schmidt deco � � c 2 | Ψ � = c n | Ψ n � A | Ψ n � B c n ≥ 0 , n = 1 n n If c 1 = 1 ⇒ | Ψ � unentangled If c i all equal ⇒ | Ψ � maximally entangled A natural measure is the entanglement entropy ( ρ A = Tr B ρ ) � c 2 n log c 2 S A ≡ − Tr ρ A log ρ A = − n = S B n Pasquale Calabrese Entanglement and CFT

Entanglement meets cond-mat and StatPhys | Ψ � is the ground state of a local Hamiltonian H Is entanglement special? Yes, if A corresponds to a spatial subset (Area law) Pasquale Calabrese Entanglement and CFT

Entanglement meets cond-mat and StatPhys | Ψ � is the ground state of a local Hamiltonian H Is entanglement special? Yes, if A corresponds to a spatial subset (Area law) In a 1+1D CFT Holzhey, Larsen, Wilczek ’94 S A = c 3 ln ℓ This is the most effective way to measure the central charge c Pasquale Calabrese Entanglement and CFT

Path integral and Riemann surfaces PC and J Cardy ’04 � Φ 1 ( x ) | ρ A | Φ 2 ( x ) � = Pasquale Calabrese Entanglement and CFT

Path integral and Riemann surfaces PC and J Cardy ’04 � Φ 1 ( x ) | ρ A | Φ 2 ( x ) � = Tr ρ n A = Tr ρ n A = for n integer is the partition function on a n -sheeted Riemann surface R n , 1 ∂ ∂ n Tr ρ n Replica trick: S A = − lim A n → 1 Pasquale Calabrese Entanglement and CFT

Riemann surfaces and CFT PC and J Cardy ’04 The Riemann surface R n , 1 is topological equivalent to the complex plane on which is mapped by � 1 / n � w → ζ = w − u w − v ; ζ → z = ζ 1 / n ⇒ w → z = w − u w − v = c n | u − v | − c 6 ( n − 1 / n ) Tr ρ n A = | u − v | = ℓ ∂ A = c ∂ n Tr ρ n ⇒ S A = − lim 3 log ℓ n → 1 Pasquale Calabrese Entanglement and CFT

More difficult problem A = Disconnected regions: Pasquale Calabrese Entanglement and CFT

More difficult problem A = Disconnected regions: More complex Riemann surface: R n , 2 of genus ( n − 1) [ R n , N has genus ( n − 1)( N − 1)] A , S A ? Tr ρ n Pasquale Calabrese Entanglement and CFT

Disjoint intervals: History A = [ u 1 , v 1 ] ∪ [ u 2 , v 2 ] In 2004 we obtained « c 6 ( n − 1 / n ) „ | u 1 − u 2 || v 1 − v 2 | Tr ρ n A = c 2 n | u 1 − v 1 || u 2 − v 2 || u 1 − v 2 || u 2 − v 1 | Tested for free fermions in different ways Casini-Huerta, Florio et al. Pasquale Calabrese Entanglement and CFT

Disjoint intervals: History A = [ u 1 , v 1 ] ∪ [ u 2 , v 2 ] In 2004 we obtained « c 6 ( n − 1 / n ) „ | u 1 − u 2 || v 1 − v 2 | Tr ρ n A = c 2 n | u 1 − v 1 || u 2 − v 2 || u 1 − v 2 || u 2 − v 1 | Tested for free fermions in different ways Casini-Huerta, Florio et al. For more complicated theories in 2008 Furukawa-Pasquier-Shiraishi and Caraglio-Gliozzi showed that it is incorrect! « c 6 ( n − 1 / n ) „ | u 1 − u 2 || v 1 − v 2 | Tr ρ n A = c 2 F n ( x ) n | u 1 − v 1 || u 2 − v 2 || u 1 − v 2 || u 2 − v 1 | x = ( u 1 − v 1 )( u 2 − v 2 ) ( u 1 − u 2 )( v 1 − v 2 ) = 4 − point ratio Pasquale Calabrese Entanglement and CFT

Compactified boson (Luttinger) Furukawa Pasquier Shiraishi � 4 F 2 ( x ) = θ 3 ( ητ ) θ 3 ( τ/η ) � θ 2 ( τ ) η ∝ R 2 , x = [ θ 3 ( τ )] 2 θ 3 ( τ ) Compared against exact diagonalization in XXZ chain Pasquale Calabrese Entanglement and CFT

Compactified boson PC Cardy Tonni ’09 � � � � F n ( x ) = Θ 0 | η Γ Θ 0 | Γ /η Using old results of CFT � � ] 2 on orbifolds Dixon et al 86 [Θ 0 | Γ Γ is an ( n − 1) × ( n − 1) matrix n − 1 2 i „ π k « » 2 π k – X Γ rs = sin β k n cos n ( r − s ) n n k = 1 β y = H y (1 − x ) , H y ( x ) = 2 F 1 ( y , 1 − y ; 1; x ) with H y ( x ) X ˆ ˜ Riemann theta function Θ( z | Γ) ≡ exp i π m · Γ · m + 2 π im · z m ∈ Z n − 1 Pasquale Calabrese Entanglement and CFT

Compactified boson PC Cardy Tonni ’09 � � � � F n ( x ) = Θ 0 | η Γ Θ 0 | Γ /η Using old results of CFT � � ] 2 on orbifolds Dixon et al 86 [Θ 0 | Γ Γ is an ( n − 1) × ( n − 1) matrix n − 1 2 i „ π k « » 2 π k – X Γ rs = sin β k n cos n ( r − s ) n n k = 1 β y = H y (1 − x ) , H y ( x ) = 2 F 1 ( y , 1 − y ; 1; x ) with H y ( x ) X ˆ ˜ Riemann theta function Θ( z | Γ) ≡ exp i π m · Γ · m + 2 π im · z m ∈ Z n − 1 • F n ( x ) invariant under x → 1 − x and η → 1 /η • We are unable to analytic continue to real n for general x and η • Only for η ≪ 1 and for x ≪ 1 Pasquale Calabrese Entanglement and CFT

Compactified boson II PC Cardy Tonni η ≪ 1 1 ( x ) = 1 2 ln η − D ′ 1 ( x ) + D ′ 1 (1 − x ) − F ′ 2 with Z i ∞ π z dz D ′ 1 ( x ) = − sin 2 π z log H z ( x ) i − i ∞ Pasquale Calabrese Entanglement and CFT

Compactified boson III PC Cardy Tonni x ≪ 1 � x � x � α � 2 α P (2) F n ( x ) = 1+2 n P n +2 n n + · · · α = min ( η, 1 /η ) 4 n 2 4 n 2 n − 1 n − 1 [sin ( π l / n )] 2 α = 1 l / n 1 X X P n = [sin ( π l / n )] 2 α 2 l =1 l =1 1 ( x ) = 2 1 − 2 α x α P ′ − F ′ 1 + . . . Pasquale Calabrese Entanglement and CFT

Compactified boson III PC Cardy Tonni x ≪ 1 � x � x � α � 2 α P (2) F n ( x ) = 1+2 n P n +2 n n + · · · α = min ( η, 1 /η ) 4 n 2 4 n 2 n − 1 n − 1 [sin ( π l / n )] 2 α = 1 l / n 1 X X P n = [sin ( π l / n )] 2 α 2 l =1 l =1 1 ( x ) = 2 1 − 2 α x α P ′ − F ′ 1 + . . . √ π Γ( α + 1) NEW P ′ 1 = α + 3 � � 4Γ 2 Pasquale Calabrese Entanglement and CFT

Compactified boson III PC Cardy Tonni x ≪ 1 � x � x � α � 2 α P (2) F n ( x ) = 1+2 n P n +2 n n + · · · α = min ( η, 1 /η ) 4 n 2 4 n 2 n − 1 n − 1 [sin ( π l / n )] 2 α = 1 l / n 1 X X P n = [sin ( π l / n )] 2 α 2 l =1 l =1 1 ( x ) = 2 1 − 2 α x α P ′ − F ′ 1 + . . . √ π Γ( α + 1) NEW P ′ 1 = α + 3 � � 4Γ 2 n − 1 L − 2 L − ℓ 1 − 1 = n NEW 2 h i P (2) X X X Q 2 α + 2 Q 2 α n 1 2 2 L =3 ℓ 1 =1 ℓ 2 =1 sin( π ( L − ℓ 1 ) / n ) sin( π ( L − ℓ 2 ) / n ) Q 1 ≡ sin( πℓ 1 / n ) sin( πℓ 2 / n ) sin( π L / n ) sin( π ( L − ℓ 1 − ℓ 2 ) / n ) sin( πℓ 1 / n ) sin( πℓ 2 / n ) Q 2 , ≡ sin( π ( L − ℓ 1 ) / n ) sin( π ( L − ℓ 2 ) / n ) sin( π L / n ) sin( π ( L − ℓ 1 − ℓ 2 ) / n ) P ( m ) have an OPE interpretation (for m = 1 [Headrick ’10]) n Pasquale Calabrese Entanglement and CFT

The XX model Fagotti PC ’10 The RDM of two intervals is not trivial because of JW string Igloi-Peschel n ( x ) = F CFT ( x ) + ( − ) ℓ ℓ − δ n f n ( x ) + . . . F lat CFT OK and δ n = 2 / n n Pasquale Calabrese Entanglement and CFT

The XX model with Open BC Fagotti PC ’10 F n ( x ) = 1 because it is free fermions! Pasquale Calabrese Entanglement and CFT

The Ising model Alba Tagliacozzo PC ’09 Monte Carlo for 2D and TTN for 1D Tr ρ 2 A MC Tr ρ 2 A TTN Large monotonic corrections to the scaling! FSS analysis confirms: " „ (1 + √ x )(1 + √ 1 − x ) # 1 / 2 « 1 / 2 1 + x 1 / 4 +((1 − x ) x ) 1 / 4 +(1 − x ) 1 / 4 F 2 ( x ) = √ 2 2 Pasquale Calabrese Entanglement and CFT

The Ising model Alba Tagliacozzo PC ’09 Von Neumann Pasquale Calabrese Entanglement and CFT

The Ising model Fagotti PC ’10 δ n = 1 / n because of Ising fermion! Pasquale Calabrese Entanglement and CFT