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The entropy of disconnected regions • Consider again a quantum spin chain which we now divide into four different regions (periodic boundary conditions) 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 ... x s x s ... x s x ... ... x s x s j+1 x ... x s x ... i i+1 x i+L−1 j j+M−1 D A : L sites C B : M sites D • A problem of current interest is finding the entanglement entropy of the region A ∪ B with respect to the rest of the system (where A and B are now “disconnected”) S A∪B = −Tr A∪B (ρ A∪B ln(ρ A∪B )) with ρ A∪B = Tr C∪D (|gs〉〈gs|) and |gs〉 is again the ground state of the chain (a pure state).
The entropy of disconnected regions and the replica trick • We can use the “replica trick” for evaluating the entanglement entropy as before: S A∪B = −Tr A∪B (ρ A∪B log(ρ A∪B )) = − lim n→1 d dn Tr A∪B(ρ n A∪B) • For integer numbers n of replicas, in the scaling limit, this is again a partition function on a (more complicated) Riemann surface: A∪B〈φ|ρ A∪B |ψ〉 A∪B ∼ Tr A∪B (ρ n A∪B) ∼ Z n = ∫ [dϕ] Mn exp [ ∫ ] − d 2 x L[ϕ](x) M n M n : n 2 . . 1 A B
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 10 and 11: Partition functions on multi-sheete
Page 12 and 13: • Another twist field ˜T is asso
Page 14 and 15: Hence we have Short- and large-dist
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

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