3.4 Reduced density matrix and entropy

The density matrix 1 2 ρ(ψ +,ψ+ ) defined by tracing over ψ−-modes is given by the Euclidean path integral over field configurations on the complete instanton (0 < ϕ < 2π ) with a cut along the axis ϕ = 0 where the field ψ(X ) in the path integral takes the values ψ1+ (x ) and ψ2+ (x) below and above the cut respectively. The trace Trρ is obtained by equating the fields across the cut and doing the unrestricted Euclidean path integral on the complete Euclidean instanton E. Analogously, n Trρ is given by the path integral over field configurations defined on the n-fold cover En of the complete instanton. This space is described by the metric (50View Equation) where angular coordinate ϕ is periodic with period 2πn. It has a conical singularity on the surface Σ so that in a small vicinity of Σ the total space E n is a direct product of Σ and a two-dimensional cone 𝒞 n with angle deficit δ = 2π (1 − n). Due to the abelian isometry generated by the Killing vector ∂ ϕ this construction can be analytically continued to arbitrary (non-integer) n → α. So that one can define a partition function
Z(α ) = Trρα (54 )
by the path integral over field configurations over E α, the α-fold cover of the instanton E. For a bosonic field described by the field operator ˆ𝒟 one has that − 1∕2 Z (α) = det ˆ𝒟. Defining the effective action as W (α ) = − ln Z(α ), the entanglement entropy is still given by formula (16View Equation), i.e., by differentiating the effective action with respect to the angle deficit. Clearly, only the term linear in (1 − α) contributes to the entropy. Thus, the problem reduces to the calculation of this term in the effective action.
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