entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...
entanglement entropy of disconnected regions - Staff.city.ac.uk - City ... entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...
Partition functions on multi-sheeted Riemann surfaces [Callan, Wilczek 1994; Holzhey, Larsen, Wilczek 1994] • We can use the “replica trick” for evaluating the entanglement entropy: S A = −Tr A (ρ A log(ρ A )) = − lim n→1 d dn Tr A(ρ n A) • For integer numbers n of replicas, in the scaling limit, this is a partition function on a Riemann surface: A〈φ|ρ A |ψ〉 A ∼ Tr A (ρ n A) ∼ Z n = ∫ [dϕ] Mn exp | ψ > < φ| A r [ − ∫ ] d 2 x L[ϕ](x) M n M n :
Branch-point twist fields [J.L. Cardy, O.C.A, B. Doyon 2007] • Consider many copies of the QFT model on the usual R 2 : L (n) [ϕ 1 , . . . , ϕ n ](x) = L[ϕ 1 ](x) + . . . + L[ϕ n ](x) • There is an obvious symmetry under cyclic exchange of the copies: L (n) [σϕ 1 , . . . , σϕ n ] = L (n) [ϕ 1 , . . . , ϕ n ] , with σϕ i = ϕ i+1 mod n • Whenever we have an internal symmetry in a QFT we can associate a twist field to it. We will call the fields associated to the Z n symmetry introduced here branch point twist fields.
- Page 1: Entanglement entropy in 1+1 dimensi
- Page 4 and 5: How and why to measure (or quantify
- Page 6 and 7: - The maximally entangled state |gs
- Page 8 and 9: Short- and large-distance entanglem
- Page 12 and 13: • Another twist field ˜T is asso
- Page 14 and 15: Hence we have Short- and large-dist
- Page 16 and 17: The entropy of disconnected regions
- Page 18 and 19: The entropy of disconnected regions
- Page 20 and 21: Some simple limit cases • When r
- Page 22 and 23: Non-extensivity • In recent years
Branch-point twist fields<br />
[J.L. Cardy, O.C.A, B. Doyon 2007]<br />
• Consider many copies <strong>of</strong> the QFT model on the usual R 2 :<br />
L (n) [ϕ 1 , . . . , ϕ n ](x) = L[ϕ 1 ](x) + . . . + L[ϕ n ](x)<br />
• There is an obvious symmetry under cyclic exchange <strong>of</strong> the copies:<br />
L (n) [σϕ 1 , . . . , σϕ n ] = L (n) [ϕ 1 , . . . , ϕ n ] , with σϕ i = ϕ i+1 mod n<br />
• Whenever we have an internal symmetry in a QFT we can associate a twist field to it. We<br />
will call the fields associated to the Z n symmetry introduced here branch point twist<br />
fields.
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