Pairing theory of the symmetry energy

A model is investigated that displays a picture of the symmetry energy as an energy of rotation in isospace of a Cooper pair condensate, briefly superfluid isorotation. The Hamiltonian is isobarically invariant and has a one- and a two-nucleon term, where the two-nucleon interaction is composed of an isovector pairing force and an interaction of isospins. It is analyzed in the Hartree-Bogolyubov plus random-phase approximation (RPA). The Hartree-Bogolyubov energy minus Lagrangian multiplier terms proportional to the number of valence nucleons and the z component of the isospin is shown to be locally minimized by a product of neutron and proton Bardeen-Cooper-Schrieffer states. The equations of the RPA can be reduced to independent equations for two-neutron, two-proton, and neutron-proton quasiparticle pairs. In each of these spaces, they have a Nambu-Goldstone solution due to the global gauge invariance and isobaric invariance of the Hamiltonian. Except for the Nambu-Goldstone solutions, the RPA solutions are independent of the strength of the isospin interaction. If, in one space, the pertinent single-nucleon spectrum has a particle-hole symmetry, the RPA solutions are twofold degenerate except for the Nambu-Goldstone solution and one more solution. In an idealized case of infinitely many equidistant single-nucleon levels, the one- nucleon term in the Hamiltonian and the isospin interaction contribute terms in the symmetry energy quadratic in the isospin T. The pairing force and the two-neutron and two-proton RPA correlation energies do not contribute. The contribution of the neutron-proton correlation energy is dominated by the Nambu-Goldstone solution, which gives a linear term that makes the total symmetry energy proportional to T (T + 1). The rest of this contribution is negative and can be written as the difference of two terms of the form root(aT)(2) + b(2) - b. Observations reported from Skyrme force calculations are discussed in the light of these results. Calculations with deformed Woods-Saxon single-nucleon levels give results similar to those of the idealized case. In calculations for the mass numbers A = 56 and A = 100 with spherical Woods-Saxon levels, the promotion of nucleons across magic gaps in the single-nucleon spectrum and the onset of superfluidity with the departure from magicity give rise to large linear terms in the symmetry energy. The calculations with Woods-Saxon single-nucleon levels reproduce surprisingly well the empirical symmetry energy. An experimental signature of superfluid isorotation is discussed.