Long-range entanglement in the Dirac vacuum

Abstract

Recently, there have been a number of works investigating the entanglement properties of distinct noncomplementary parts of discrete and continuous bosonic systems in ground and thermal states. The relativistic fermionic case, however, has yet to be expressly addressed. In this paper we investigate the entanglement between a pair of far-apart regions of the $(3+1)$-dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of $L∕R$, where $L$ is the distance between the regions and $R$ is their typical scale, it decays no faster than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.