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

Noburo Shiba

doi:10.1007/JHEP12(2014)152

paper

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

arXiv:1408.0637 · doi:10.1007/JHEP12(2014)152

Abstract

We develop the computational method of entanglement entropy based on the idea that $T r ρ_{Ω}^{n}$ 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 $I^{(n)} (A, B) = S_{A}^{(n)} + S_{B}^{(n)} - S_{A \cup B}^{(n)}$ 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, $I^{(n)} (A, B) - I^{(n)} (A, B) ∣_{r \to \infty} = C_{A B}^{(n)} / r^{α (d - 1)}$ where $α = 1$ or 2 which is determined by the property of the local operators under the transformation $ϕ \to - ϕ$ and $α = 2$ for the vacuum state. We give a method to compute $C_{A B}^{(2)}$ systematically.

arXiv comment22 pages; v3, typos corrected, published version

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

Abstract

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