# Entanglement Content of Quantum Particle Excitations II. Disconnected Regions and Logarithmic Negativity

## TL;DR

This paper investigates how entanglement measures, specifically entanglement entropy and logarithmic negativity, change in excited states of a free massive bosonic theory for disconnected regions, revealing region-size dependencies and polynomial structures.

## Contribution

It extends previous work by analyzing disconnected regions and deriving explicit formulas for entanglement measures in excited states, including their polynomial and functional structures.

## Key findings

- Entanglement entropy change depends only on total size of regions.
- Logarithmic negativity increment is a polynomial in region sizes.
- Results agree with numerical simulations on a harmonic chain.

## Abstract

In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement.