Entanglement entropy at quantum critical points: Can you ... - INFN

•
〈{φ A 1 }|ρ A |{φ A 2 }〉 = Tr φ B (〈{φ A 1 }| ⊗ 〈{φ B }|ψ 0 〉〈ψ 0 (|{φ A 2 }〉 ⊗ |{φ B }〉
= 1 Z c
∫
(dφ B )e −(SA (φ A 1 )/2+SA (φ A 2 )/2+S∂ (φ A 1 ,φB )/2+S ∂ (φ A 2 ,φB )/2+S B (φ B )) .
Here the action has been divided into regions A, B, and the boundary ∂, where the
last takes into account contributions mixing the A and B degrees of freedom
• Higher powers of the density matrix need not trace to unity: Tr ρ n A is now a sum
over n configurations defined in A and n configurations defined in B. The key is to
keep track of how these different configurations are stitched together at the boundary
by the terms S ∂ that link A and B: {φ A i } is linked to {φ B i } as well as {φ B i+1} for
i = 1,...,n − 1, and {φ A n } is linked to {φ B n } and {φ B 1 }. This is normalized
through division by (Z c ) n , which can be thought of again as n copies of A and B
configurations, but with {φ A i } linked only to {φ B i }.
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