entanglement entropy of disconnected regions - Staff.city.ac.uk - City ...

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