Fermionic vacua and entanglement in hyperbolic de Sitter spacetime

We point out a non-trivial feature of the vacuum structure of free massive or massless Dirac fields in the hyperbolic de Sitter spacetime. Here we have two causally disconnected regions, say $R$ and $L$ separated by another region, $C$, appropriate to discuss the long range correlations of two superhorizon separated observers. There are local modes of the Dirac field having supports individually either in $R$ or $L$ as well as global modes found via analytically continuing the $R$ modes to $L$ and vice versa. However, we show that unlike the case of a real scalar field, the analytic continuation does not preserve the orthogonality of the resulting global modes. Accordingly, we need to orthonormalise these modes by making their suitable linear combinations prior to the field quantisation, in order to preserve the canonical anti-commutation relations. The most general form of this orthonormalisation is achieved via introducing a spacetime independent continuous parameter, $\theta_{\rm RL}$, resulting in a one-parameter family of global vacua. We emphasise that unlike the case of the so called de Sitter $\alpha$-vacua, introducing such parametrisation is mandatory in our present scenario, in order to preserve the natural canonical structure. Using these vacua, we next investigate both entanglement and R\'enyi entropies by tracing over the states belonging to either $R$ or $L$ region and demonstrate their variations with respect to the parameter $\theta_{\rm RL}$.