Entanglement entropy of disjoint regions in excited states: an operator method

Abstract

Abstract: We develop the computational method of entanglement entropy based on the idea that (Formula Presented.)(Formula Presented.) is written as the expectation value of the local operator, where ρΩ is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I(n)(A, B) = SA(n) + SB(n) − SA ∪ 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 RA,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 → ∞ = CAB(n)/rα(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute CAB(2) systematically.