2.6 The Euclidean path integral representation and the replica method

A technical method very useful for the calculation of the entanglement entropy in a field theory is the the replica trick, see [33Jump To The Next Citation Point]. Here we illustrate this method for a field theory described by a second-order Laplace-type operator. One considers a quantum field ψ (X ) in a d-dimensional spacetime and chooses the Cartesian coordinates X μ = (τ,x,zi, i = 1,..,d− 2), where τ is Euclidean time, such that the surface Σ is defined by the condition x = 0 and (zi, i = 1,..,d− 2) are the coordinates on Σ. In the subspace (τ, x) it will be convenient to choose the polar coordinate system τ = rsin(ϕ) and x = rcos(ϕ), where the angular coordinate ϕ varies between 0 and 2π. We note that if the field theory in question is relativistic, then the field operator is invariant under the shifts ϕ → ϕ + w, where w is an arbitrary constant.

One first defines the vacuum state of the quantum field in question by the path integral over a half of the total Euclidean spacetime defined as τ ≤ 0 such that the quantum field satisfies the fixed boundary condition ψ(τ = 0,x, z) = ψ0(x,z) on the boundary of the half-space,

∫ Ψ[ψ0(x,z )] = 𝒟 ψ e−W [ψ ], (13 ) ψ(X )|τ=0=ψ0(x,z)
where W [ψ] is the action of the field. The surface Σ in our case is a plane and the Cartesian coordinate x is orthogonal to Σ. The co-dimension 2 surface Σ defined by the conditions x = 0 and τ = 0 naturally separates the hypersurface τ = 0 into two parts: x < 0 and x > 0. These are the two sub-regions A and B discussed in Section 2.1.

The boundary data ψ (x,z) is also separated into ψ− (x,z) = ψ0(x,z ), x < 0 and ψ+ = ψ0(x,z),x > 0. By tracing over ψ − (x,z) one defines a reduced density matrix

1 2 ∫ 1 2 ρ(ψ+,ψ +) = 𝒟 ψ − Ψ (ψ+,ψ − )Ψ (ψ+,ψ − ) , (14 )
where the path integral goes over fields defined on the whole Euclidean spacetime except a cut (τ = 0,x > 0). In the path integral the field ψ(X ) takes the boundary value ψ2+ above the cut and ψ1 + below the cut. The trace of the n-th power of the density matrix (14View Equation) is then given by the Euclidean path integral over fields defined on an n-sheeted covering of the cut spacetime. In the polar coordinates (r,ϕ) the cut corresponds to values ϕ = 2πk, k = 1,2,..,n. When one passes across the cut from one sheet to another, the fields are glued analytically. Geometrically this n-fold space is a flat cone Cn with angle deficit 2π (1 − n) at the surface Σ. Thus, we have
Trρn = Z [Cn], (15 )
where Z[C ] n is the Euclidean path integral over the n-fold cover of the Euclidean space, i.e., over the cone Cn. Assuming that in Eq. (15View Equation) one can analytically continue to non-integer values of n, one observes that
α − Trˆρ ln ˆρ = − (α∂α − 1) ln Trρ |α=1 ,
where ˆρ = ρ∕Trρ is the renormalized matrix density. Introduce the effective action W (α) = − lnZ (α), where Z(α ) = Z[Cα ] is the partition function of the field system in question on a Euclidean space with conical singularity at the surface Σ. In the polar coordinates (r,ϕ) the conical space C α is defined by making the coordinate ϕ periodic with period 2πα, where (1 − α ) is very small. The invariance under the abelian isometry ϕ → ϕ + w helps to construct without any problem the correlation functions with the required periodicity 2π α starting from the 2π-periodic correlation functions. The analytic continuation of Trρα to α different from 1 in the relativistic case is naturally provided by the path integral Z (α) over the conical space C α. The entropy is then calculated by the replica trick
S = (α ∂ − 1)W (α)| . (16 ) α α=1
One of the advantages of this method is that we do not need to care about the normalization of the reduced density matrix and can deal with a matrix, which is not properly normalized.
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