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

TL;DR

This work presents an operator-based framework to compute Rényi entanglement entropies in quantum field theories by expressing $\mathrm{Tr}\rho_{\Omega}^n$ as an expectation value of a glueing operator $E_{\Omega}$. The method, developed for a free scalar theory and extended to general QFTs with a mass gap, yields explicit expressions for mutual Rényi information in locally excited states and reveals a clean entangled-state interpretation at large separations. For massless free scalars, the authors show power-law decay of mutual information with distance, with the decay exponent set by operator parity, and they provide a systematic route to compute the coefficient $C^{(2)}_{AB}$ via a generating functional. The approach unifies vacuum and excited-state analyses, connects to Cardy’s capacitance picture, and offers a versatile toolkit for exploring entanglement in interacting theories, thermal states, and fermionic systems.

Abstract

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.