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Entanglement Entropy of Disjoint Regions in Excited States : An Operator Method

We develop the computational method of entanglement entropy based on the idea that $Trρ_Ω^n$ is written as the expectation value of the local operator, where $ρ_Ω$ is a density matrix of the subsystem $Ω$. We apply it to consider the mutual Renyi information $I^{(n)}(A,B)=S^{(n)}_A+S^{(n)}_B-S^{(n)}_{A\cup B}$ of disjoint compact spatial regions $A$ and $B$ in the locally excited states defined by acting the local operators at $A$ and $B$ on the vacuum of a $(d+1)$-dimensional field theory, in the limit when the separation $r$ between $A$ and $B$ is much greater than their sizes $R_{A,B}$. For the general QFT which has a mass gap, we compute $I^{(n)}(A,B)$ explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, $I^{(n)}(A,B)-I^{(n)}(A,B)|_{r \rightarrow \infty} =C^{(n)}_{AB}/r^{α(d-1)}$ where $α=1$ or 2 which is determined by the property of the local operators under the transformation $ϕ\rightarrow -ϕ$ and $α=2$ for the vacuum state. We give a method to compute $C^{(2)}_{AB}$ systematically.