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

TL;DR

This paper introduces a new operator-based computational method for entanglement entropy in quantum field theories, especially for disjoint regions in excited states, and provides explicit results for massive and massless fields.

Contribution

It develops an operator expectation value approach to compute entanglement entropy and mutual information for disjoint regions in excited states of QFTs, including explicit formulas for mass gap and massless cases.

Findings

Explicit computation of mutual Renyi information for massive QFTs.

Power-law decay of mutual information in massless scalar fields.

Method to systematically compute coefficients in the decay of mutual information.

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 $I^{(n)}(A,B)=S^{(n)}_A+S^{(n)}_B-S^{(n)}_{A\cup B}$ 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, $I^{(n)}(A,B)-I^{(n)}(A,B)|_{r \rightarrow \infty} =C^{(n)}_{AB}/r^{\alpha (d-1)}$ where $\alpha=1$ or 2 which is determined by the property of the local operators under the transformation $\phi \rightarrow -\phi$ and $\alpha=2$ for the vacuum state. We give a method to compute $C^{(2)}_{AB}$ systematically.