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

Abstract

We develop the computational method of entanglement entropy based on the idea that $Tr{\rho }_{\Omega }^n$ is written as the expectation value of the local operator, where ${\rho }_{\Omega }$ is a density matrix of the subsystem $\Omega$. 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, $I^{(n)}(A,B)-I^{(n)}(A,B){|}_{r\to \infty }=C_{AB}^{(n)}/r^{\alpha (d-1)}$ where $\alpha =1$ or 2 which is determined by the property of the local operators under the transformation $\varphi \to -\varphi$ and $\alpha =2$ for the vacuum state. We give a method to compute $C_{AB}^{(2)}$ systematically.

22 pages; v3, typos corrected, published version

References in corpus (5)

• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Remarks on the entanglement entropy for disconnected regions
• Entanglement of Local Operators in large N CFTs
• Notes on Quantum Entanglement of Local Operators
• Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres

Cited by in corpus (5)

• Quantum Entanglement of Localized Excited States at Finite Temperature
• Pseudo Entropy in Free Quantum Field Theories
• Aspects of Pseudo Entropy in Field Theories
• Quantum dimensions from local operator excitations in the Ising model
• Euclidean Time Approach to Entanglement Entropy on Lattices and Fuzzy Spaces