Cold dilute nuclear matter with $\alpha$-particle condensation in a generalized nonlinear relativistic mean-field model

We explore the thermodynamic properties of homogeneous cold (zero-temperature) nuclear matter including nucleons and $\alpha$-particle condensation at low densities by using a generalized nonlinear relativistic mean-field (gNL-RMF) model. In the gNL-RMF model, the $\alpha$-particle is included as explicit degree of freedom and treated as point-like particle with its interaction described by meson exchanges and the in-medium effects on the $\alpha$ binding energy is described by density- and temperature-dependent energy shift with the parameters obtained by fitting the experimental Mott density. We find that below the dropping density $n_{\rm{drop}}$ ($\sim 3\times 10^{-3}$ fm$^{-3}$), the zero-temperature symmetric nuclear matter is in the state of pure Bose-Einstein condensate (BEC) of $\alpha$ particles while the neutron-rich nuclear matter is composed of $\alpha$-BEC and neutrons. Above the $n_{\rm{drop}}$, the fraction of $\alpha$-BEC decreases with density and vanishes at the transition density $n_t$ ($\sim 8\times 10^{-3}$ fm$^{-3}$). Above the $n_t$, the nuclear matter becomes pure nucleonic matter. Our results indicate that the empirical parabolic law for the isospin asymmetry dependence of nuclear matter equation of state is heavily violated by the $\alpha$-particle condensation in the zero-temperature dilute nuclear matter, making the conventional definition of the symmetry energy meaningless. We investigate the symmetry energy defined under parabolic approximation for the zero-temperature dilute nuclear matter with $\alpha$-particle condensation, and find it is significantly enhanced compared to the case without clusters and becomes saturated at about $7$ MeV at very low densities ($\lesssim 10^{-3}$ fm$^{-3}$). The critical temperature for $\alpha$-condensation in homogeneous dilute nuclear matter is also discussed.