Universal scaling of the logarithmic negativity in massive quantum field theory

TL;DR

This paper demonstrates that the logarithmic negativity in 1+1-dimensional massive quantum field theories exhibits universal exponential decay behavior governed by the mass spectrum, enabling insights into bound states and entanglement properties.

Contribution

It provides a universal scaling law for the logarithmic negativity in massive quantum field theories, independent of scattering details, and shows how to detect bound states through negativity analysis.

Findings

Negativity saturates to a finite value for adjacent regions as r→∞.

Negativity tends to zero for separated semi-infinite regions as r→∞.

Exponential decay corrections are controlled solely by the mass spectrum.

Abstract

We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1+1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length $r$ and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance $r$. We show that the former saturates to a finite value, and that the latter tends to zero, as $r\rightarrow\infty$. We show that in both cases, the leading corrections are exponential decays in $r$ (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy, the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the entanglement entropy, a large-$r$ analysis of the negativity allows for the detection of bound states.