Equation of State of Neutron-Rich Matter in $d$-Dimensions

Nuclear systems under constraints, with high degrees of symmetries and/or collectivities may be considered as moving effectively in spaces with reduced spatial dimensions. We first derive analytical expressions for the nucleon specific energy $E_0(\rho)$, pressure $P_0(\rho)$, incompressibility coefficient $K_0(\rho)$ and skewness coefficient $J_0(\rho)$ of symmetric nucleonic matter (SNM), the quadratic symmetry energy $E_{\rm{sym}}(\rho)$, its slope parameter $L(\rho)$ and curvature coefficient $K_{\rm{sym}}(\rho)$ as well as the fourth-order symmetry energy $E_{\rm{sym,4}}(\rho)$ of neutron-rich matter in general $d$ spatial dimensions (abbreviated as "$d$D") in terms of the isoscalar and isovector parts of the isospin-dependent single-nucleon potential according to the generalized Hugenholtz-Van Hove (HVH) theorem. The equation of state (EOS) of nuclear matter in $d$D can be linked to that in the conventional 3-dimensional (3D) space by the $\epsilon$-expansion which is a perturbative approach successfully used previously in treating second-order phase transitions and related critical phenomena and more recently in studying the EOS of cold atoms. The $\epsilon$-expansion of nuclear EOS in $d$D based on a reference dimension $d_{\rm{f}}=d-\epsilon$ is shown to be effective with $-1\lesssim\epsilon\lesssim1$ starting from $1\lesssim d_{\rm{f}}\lesssim3$ in comparison with the exact expressions derived using the HVH theorem. Moreover, the EOS of SNM (with/without considering its potential part) is found to be reduced (enhanced) in lower (higher) dimensions, indicating in particular that the many-nucleon system tends to be deeper bounded but saturate at higher densities in spaces with lower dimensions. The links between the EOSs in 3D and $d$D spaces from the $\epsilon$-expansion provide new perspectives to the EOS of neutron-rich matter.