We introduce a finite-range pseudopotential built as an expansion in derivatives up to next-to-next-to-next-to-leading order (N$^3$LO) and we calculate the corresponding nonlocal energy density functional (EDF). The coupling constants of the nonlocal
EDF, for both finite nuclei and infinite nuclear matter, are expressed through the parameters of the pseudopotential. All central, spin-orbit, and tensor terms of the pseudopotential are derived both in the spherical-tensor and Cartesian representation. At next-to-leading order (NLO), we also derive relations between the nonlocal EDF expressed in the spherical-tensor and Cartesian formalism. Finally, a simplified version of the finite-range pseudopotential is considered, which generates the EDF identical to that generated by a local potential.
Background: The next-to-next-to-next-to-leading order (N3LO) nuclear energy density functional extends the standard Skyrme functional with new terms depending on higher-order derivatives of densities, introduced to gain better precision in the nuclea
r many-body calculations. A thorough study of the transformation properties of the functional with respect to different symmetries is required, as a step preliminary to the adjustment of the coupling constants. Purpose: Determine to which extent the presence of higher-order derivatives in the functional can be compatible with the continuity equation. In particular, to study the relations between the validity of the continuity equation and invariance of the functional under gauge transformations. Methods: Derive conditions for the validity of the continuity equation in the framework of time-dependent density functional theory. The conditions apply separately to the four spin-isospin channels of the one-body density matrix. Results: We obtained four sets of constraints on the coupling constants of the N3LO energy density functional that guarantee the validity of the continuity equation in all spin-isospin channels. In particular, for the scalar-isoscalar channel, the constraints are the same as those resulting from imposing the standard U(1) local-gauge-invariance conditions. Conclusions: Validity of the continuity equation in the four spin-isospin channels is equivalent to the local-gauge invariance of the energy density functional. For vector and isovector channels, such validity requires the invariance of the functional under local rotations in the spin and isospin spaces.
We study the linear response of the inner crust of neutron stars within the Random Phase Approximation, employing a Skyrme-type interaction as effective interaction. We adopt the Wigner-Seitz approximation, and consider a single unit cell of the Coul
omb lattice which constitutes the inner crust, with a nucleus at its center, surrounded by a sea of free neutrons. With the use of an appropriate operator, it is possible to analyze in detail the properties of the vibrations of the surface of the nucleus and their interaction with the modes of the sea of free neutrons, and to investigate the role of shell effects and of resonant states.
The 1S0 pairing gap associated with the inner crust of a neutron star is calculated, taking into account the coexistence of the nuclear lattice with the sea of free neutrons (finite size effects), as well as medium polarization effects associated wit
h the exchange of density and spin fluctuations. Both effects are found to be important and to lead to an overall quenching of the pairing gap. This result, whose quantitative value is dependent on the effective interaction used to generate the single-particle levels, is a consequence of the balance between the attractive (repulsive) induced interaction arising from the exchange of density (spin) modes, balance which in turn is influenced by the presence of the protons and depends on the single-particle structure of the system.
