Entanglement content of quantum particle excitations. III. Graph partition functions

Abstract

We consider two measures of entanglement, the logarithmic negativity, and the entanglement entropy, between regions of space in excited states of many-body systems formed by a finite number of particle excitations. In Papers I and II [O. A. Castro-Alvaredo et al., J. High Energy Phys. 2018(10), 39; ibid., e-print arXiv:1904.01035 (2019)], it has been shown in one-dimensional free-particle models that, in the limit of large system's and regions' sizes, the contribution from the particles is given by the entanglement of natural qubit states, representing the uniform distribution of particles in space. We show that the replica logarithmic negativity and Rényi entanglement entropy of such qubit states are equal to the partition functions of certain graphs, formed out of Wick pairings, which encode the topology of the manifold induced by permutation twist fields. Using this new connection, we provide a general proof, in the massive free boson model, which the qubit result holds in any dimensionality and for any regions' shapes and topology. The proof is based on clustering and the permutation-twist exchange relations and is potentially generalizable to other situations, such as lattice models, particle and hole excitations above generalized Gibbs ensembles, and interacting integrable models.