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We report a non-trivial feature of the vacuum structure of free massive ormassless Dirac fields in the hyperbolic de Sitter spacetime. Here we have twocausally disconnected regions, say $R$ and $L$ separated by another region,$C$. We are interested in the field theory in $R\cup L$ to understand the longrange quantum correlations between $R$ and $L$. There are local modes of theDirac field having supports individually either in $R$ or $L$, as well asglobal modes found via analytically continuing the $R$ modes to $L$ and viceversa. However, we show that unlike the case of a scalar field, the analyticcontinuation does not preserve the orthogonality of the resulting global modes.Accordingly, we need to orthonormalise them following the Gram-Schmidtprescription, prior to the field quantisation in order to preserve thecanonical anti-commutation relations. We observe that this prescriptionnaturally incorporates a spacetime independent continuous parameter,$\theta_{\rm RL}$, into the picture. Thus interestingly, we obtain a naturallyemerging one-parameter family of $\alpha$-like de Sitter vacua. The values of$\theta_{\rm RL}$ yielding the usual thermal spectra of massless createdparticles are pointed out. Next, using these vacua, we investigate bothentanglement and R\'enyi entropies of either of the regions and demonstratetheir dependence on $\theta_{\rm RL}$.

Emergent $\alpha$-like fermionic vacuum structure and entanglement in the hyperbolic de Sitter spacetime