New results on block entanglement in 1D systems Pasquale Calabrese - PowerPoint PPT Presentation

New results on block entanglement in 1D systems Pasquale Calabrese Dipartimento di Fisica Universit` a di Pisa Florence September 2008 With J. Cardy, M, Campostrini & B. Nienhuis, A. Lefevre Pasquale Calabrese Entanglement in 1D systems

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 in 1D systems

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 in 1D systems

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 in 1D systems

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 � c 2 n log c 2 S A = − n = S B n S A = 0 when | Ψ � is unentangled and its maximal = log dim H min A , B when c n are equals Pasquale Calabrese Entanglement in 1D systems

Entanglement meets cond-mat (and QFT) | Ψ � is the ground state of a local Hamiltonian H Is entanglement special? Pasquale Calabrese Entanglement in 1D systems

Entanglement meets cond-mat (and QFT) | Ψ � is the ground state of a local Hamiltonian H Is entanglement special? Yes, if A is a large compact spatial subset How does S A depend on the size of A ? What about the shape of A? Is there any universality? Pasquale Calabrese Entanglement in 1D systems

Area law and criticality Area Law: S A ∝ A [Non extensive] Srednicki ’93 ↓ (lots of works) ↓ Wolf et al ’07 Only in gapped systems Pasquale Calabrese Entanglement in 1D systems

Area law and criticality Area Law: S A ∝ A [Non extensive] Srednicki ’93 ↓ (lots of works) ↓ Wolf et al ’07 Only in gapped systems Holzhey, Larsen, Wilczek ’94: In a 1+1D T = 0 CFT S A = c 3 ln ℓ a Pasquale Calabrese Entanglement in 1D systems

Area law and criticality Area Law: S A ∝ A [Non extensive] Srednicki ’93 ↓ (lots of works) ↓ Wolf et al ’07 Only in gapped systems Holzhey, Larsen, Wilczek ’94: In a 1+1D T = 0 CFT S A = c 3 ln ℓ a Vidal, Latorre, Rico, Kitaev ’03: QI perspective Pasquale Calabrese Entanglement in 1D systems

Area law and criticality Area Law: S A ∝ A [Non extensive] Srednicki ’93 ↓ (lots of works) ↓ Wolf et al ’07 Only in gapped systems Holzhey, Larsen, Wilczek ’94: In a 1+1D T = 0 CFT S A = c 3 ln ℓ a Vidal, Latorre, Rico, Kitaev ’03: QI perspective Extensive reviews by Amico et al., Eisert et al. [RMP] Pasquale Calabrese Entanglement in 1D systems

Entanglement and CFT (with J. Cardy) ∂ n Replica trick: S A = − Tr ρ A log ρ A = − lim ∂ n Tr ρ A n → 1 For n integer, Tr ρ n A is a partition function ⇒ analytic calcs are possible! Pasquale Calabrese Entanglement in 1D systems

Entanglement and CFT (with J. Cardy) ∂ n Replica trick: S A = − Tr ρ A log ρ A = − lim ∂ n Tr ρ A n → 1 For n integer, Tr ρ n A is a partition function ⇒ analytic calcs are possible! In CFT, Tr ρ n A transforms like the correlation function of m (# of points between A & B) primary fields with scaling dimension � ℓ � − c 6 ( n − 1 n ) S A = c 3 ln ℓ c � n − 1 � Tr ρ n ∆ Φ = ⇒ A = c n ⇒ a + c ′ 1 24 n a Pasquale Calabrese Entanglement in 1D systems

Entanglement and CFT (with J. Cardy) ∂ n Replica trick: S A = − Tr ρ A log ρ A = − lim ∂ n Tr ρ A n → 1 For n integer, Tr ρ n A is a partition function ⇒ analytic calcs are possible! In CFT, Tr ρ n A transforms like the correlation function of m (# of points between A & B) primary fields with scaling dimension � ℓ � − c 6 ( n − 1 n ) S A = c 3 ln ℓ c � n − 1 � Tr ρ n ∆ Φ = ⇒ A = c n ⇒ a + c ′ 1 24 n a Finite temperature π c ℓ 8 β , ℓ ≫ β classical extensive „ β > > 3 « > S A = c π a sinh πℓ < + c ′ 3 log 1 ≃ β 3 log ℓ c > > > a , ℓ ≪ β T = 0 non − extensive : Pasquale Calabrese Entanglement in 1D systems

Entanglement and CFT (with J. Cardy) ∂ n Replica trick: S A = − Tr ρ A log ρ A = − lim ∂ n Tr ρ A n → 1 For n integer, Tr ρ n A is a partition function ⇒ analytic calcs are possible! In CFT, Tr ρ n A transforms like the correlation function of m (# of points between A & B) primary fields with scaling dimension � ℓ � − c 6 ( n − 1 n ) S A = c 3 ln ℓ c � n − 1 � Tr ρ n ∆ Φ = ⇒ A = c n ⇒ a + c ′ 1 24 n a Finite temperature π c ℓ 8 β , ℓ ≫ β classical extensive „ β > > 3 « > S A = c π a sinh πℓ < + c ′ 3 log 1 ≃ β c 3 log ℓ > > > a , ℓ ≪ β T = 0 non − extensive : Finite size „ L « S A = c π a sin πℓ + c ′ 3 log Symmetric ℓ → L − ℓ . Maximal for ℓ = L / 2 1 L Pasquale Calabrese Entanglement in 1D systems

Open systems 12 ( n − 1 � c n ) � 2 ℓ 6 log 2 ℓ ⇒ S A = c Tr ρ An = ˜ c ′ c n a + ˜ 1 a Pasquale Calabrese Entanglement in 1D systems

Open systems 12 ( n − 1 � c n ) � 2 ℓ 6 log 2 ℓ ⇒ S A = c Tr ρ An = ˜ c ′ c n a + ˜ 1 a finite temperature „ β π a sinh 2 πℓ « S A ( β ) = c c ′ 6 log + ˜ 1 β and finite size „ 2 L S A ( L ) = c π a sin πℓ « c ′ 6 log + ˜ 1 L c ′ 1 − c ′ ˜ 1 / 2 = ln g boundary entropy [Affleck, Ludwig] Pasquale Calabrese Entanglement in 1D systems

Open systems 12 ( n − 1 � c n ) � 2 ℓ 6 log 2 ℓ ⇒ S A = c Tr ρ An = ˜ c ′ c n a + ˜ 1 a finite temperature „ β π a sinh 2 πℓ « S A ( β ) = c c ′ 6 log + ˜ 1 β and finite size „ 2 L S A ( L ) = c π a sin πℓ « c ′ 6 log + ˜ 1 L c ′ 1 − c ′ ˜ 1 / 2 = ln g boundary entropy [Affleck, Ludwig] [From Laflorencie et al ’06] Pasquale Calabrese Entanglement in 1D systems

Developments Since the early papers in 2003 about 1000 papers on the subject! Pasquale Calabrese Entanglement in 1D systems

Developments Since the early papers in 2003 about 1000 papers on the subject! Effective way of detecting and characterizing quantum criticality In random (no conformal invariance!) quantum spin chains S A ∝ log ℓ Rafael and Moore, Laflorencie, Santachiara. . . It is related to the number of broken singlets. Is it true for clean chains? NO Alet et al, Jacobsen and Saleur Topological entanglement entropy S A = α L − γ, γ is the topological charge Kitaev and Preskill, Levin and Wen, Fradkin and Moore, Schoutens et al., Furukawa and Misguich, Li and Haldane. . . New numerical methods based on entanglement to simulate d > 1 Vidal, Latorre, Cirac, Hastings . . . . . . . . . Time dependence and DMRG-like simulability of non-equilibrium PC and JC, Vidal, Schollwoeck, Kollath, Eisert, Cirac, Hastings, Peschel . . . . . . . . . Holography: S A = length of the geodesic in the AdS bulk Ryu and Takayanagi. . . Too many more, sorry if YOUR name is not here! Pasquale Calabrese Entanglement in 1D systems

Universal finite size scaling in Heisenberg chains Joint work with B. Nienhuis and M. Campostrini L � j +1 + σ y j σ y [ σ x j σ x j +1 − ∆ σ z j σ z H = − j +1 ] j =1 with periodic boundary conditions Pasquale Calabrese Entanglement in 1D systems

Universal finite size scaling in Heisenberg chains Joint work with B. Nienhuis and M. Campostrini L � j +1 + σ y j σ y [ σ x j σ x j +1 − ∆ σ z j σ z H = − j +1 ] j =1 with periodic boundary conditions 1 − 1 ≤ ∆ ≤ 1: gapless 2 ∆ = 0: free fermions 3 ∆ = − 1 / 2 with L odd: magic Doubly degenerate ground state with no FS for the energy E 0 = − 3 / 2 L exactly Baxter The components of the ground-state wavefunction (suitable normalized) are integer numbers related to the combinatorics of Alternating Sign Matrices, Plane partitions etc Razumov-Stroganov Correlations are simple functions (rational/factorial) of L Pasquale Calabrese Entanglement in 1D systems