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Universal scaling of the logarithmic negativity in massive quantum field theory

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.