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Progress of Theoretical Physics

Necessary Conditions on Spin-Isospin-Dependent Liquid Structure Functions of Fermion Systems

Masatoshi Takano, Masumi Yamada

Published Dec 1, 1992• 2 citations
Publisher

Abstract

Spin-isospin-dependent liquid structure functions are defined in terms of the projection operators onto particular spin-isospin pair states. Then, necessary conditions on these structure functions are derived in the form of inequalities. The systems studied are liquid 3He, neutron matter, symmetric nuclear matter, asymmetric nuclear matter, and partialy spin-polarized asymmetric nuclear matter. More than one inequalities is obtained for each of these systems, and the number of the inequalities increases with increase of the spin-isospin degree of freedom. It is argued that these conditions can be used to find approximate kinetic-energy expressions of fermion systems.

DOI
10.1143/ptp.88.1131

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\nWe discuss weakly bound states of a few-fermion system having spin-isospin symmetry. This corresponds to the nuclear physics case in which the singlet, a0, and triplet, a1, n − p scattering lengths are large with respect to the range of the nuclear interaction. The ratio of the two is about a0/a1 ≈ −4.31. This value defines a plane in which a0 and a1 can be varied up to the unitary limit, 1/a0 = 0 and 1/a1 = 0, maintaining its ratio fixed. Using a spin dependant potential model we estimate the three-nucleon binding energy along that plane. This analysis can be considered an extension of the Efimov plot for three bosons to the case of three 1/2-spin-isospin fermions.\n

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Spin-isospin-dependent liquid structure functions are defined in terms of the projection operators onto particular spin-isospin pair states. Then, necessary conditions on these structure functions are derived in the form of inequalities. The systems studied are liquid 3He, neutron matter, symmetric nuclear matter, asymmetric nuclear matter, and partialy spin-polarized asymmetric nuclear matter. More than one inequalities is obtained for each of these systems, and the number of the inequalities increases with increase of the spin-isospin degree of freedom. It is argued that these conditions can be used to find approximate kinetic-energy expressions of fermion systems.
This paper presents the results of investigation of the superfluid states in a two-component Fermi liquid within the framework of the Fermi liquid approach. Particular attention is paid to the superfluid states in nuclear matter which are characterized by the superposition of singlet and triplet types of pairing in spin and isospin spaces. The fundamental principles of the Fermi liquid approach are formulated and further used in the study of superfluidity in nuclear matter with the superposition of singlet and triplet types of pairing. The system of self-consistency equations is derived and its solutions are presented. The conditions of existence of the considered states are determined. These conditions impose certain constraints on the interaction potential and particle density in the system. It is shown that the states with a complete set of nonzero order parameters are realized only in a narrow density range, the width and position of which on the density scale depend on the actual parametrization of the Skyrme potential employed. We considered 18 different parameterizations and indicated for which of them the studied types of superfluid states may appear. The problem of stability of the states with the superposition of singlet and triplet pairing is studied. It is shown that the lowest value of the thermodynamic potential corresponds to purely triplet states. Then, in the order of increasing, follow the thermodynamic potentials of purely singlet states and mixed singlet-triplet states. Separately, we consider the case of unitary states, for which the solutions of the self-consistency equations are also analyzed. It is shown that the density range in which such states can be realized is different from that corresponding to the nonunitary states. The problem of existence of the unitary superfluid states with the superposition of singlet and triplet superfluidity is considered in the case of asymmetrical nuclear matter. It is shown that the appearance of asymmetry in nuclear matter breaks the unitarity of superfluid states in nuclear matter with regard to isospin.
The possibility of appearance of spin polarized states in symmetric and strongly asymmetric nuclear matter is analyzed within the framework of a Fermi liquid theory with the Skyrme effective interaction. The zero temperature dependence of the neutron and proton spin polarization parameters as functions of density is found for SkM∗^*∗, SGII (symmetric case) and SLy4, SLy5 (strongly asymmetric case) effective forces. By comparing free energy densities, it is shown that in symmetric nuclear matter ferromagnetic spin state (parallel orientation of neutron and proton spins) is more preferable than antiferromagnetic one (antiparallel orientation of spins). Strongly asymmetric nuclear matter undergoes at some critical density a phase transition to the state with the oppositely directed spins of neutrons and protons while the state with the same direction of spins does not appear. In comparison with neutron matter, even small admixture of protons strongly decreases the threshold density of spin instability. It is clarified that protons become totally polarized within a very narrow density domain while the density profile of the neutron spin polarization parameter is characterized by the appearance of long tails near the transition density.
Spin polarized states in nuclear matter with an effective nucleon-nucleon interaction are studied for a wide range of isospin asymmetries and densities. Based on a Fermi liquid theory, it is shown that there are a few possible scenarios of spin ordered phase transitions: (a) nuclear matter undergoes at some critical density a phase transition to a spin polarized state with the oppositely directed spins of neutrons and protons (Skyrme SLy4 and Gogny D1S interactions); (b) at some critical density, a spin polarized state with the like-directed neutron and proton spins appears (Skyrme SkI5 interaction); (c) nuclear matter under increasing density, at first, undergoes a phase transition to the state with the opposite directions of neutron and proton spins, which goes over at larger density to the state with the same direction of nucleon spins (Skyrme SkI3 interaction).
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We discuss weakly bound states of a few-fermion system having spin-isospin symmetry. This corresponds to the nuclear physics case in which the singlet, a0, and triplet, a1, n − p scattering lengths are large with respect to the range of the nuclear interaction. The ratio of the two is about a0/a1 ≈ −4.31. This value defines a plane in which a0 and a1 can be varied up to the unitary limit, 1/a0 = 0 and 1/a1 = 0, maintaining its ratio fixed. Using a spin dependant potential model we estimate the three-nucleon binding energy along that plane. This analysis can be considered an extension of the Efimov plot for three bosons to the case of three 1/2-spin-isospin fermions.
\nWe discuss weakly bound states of a few-fermion system having spin-isospin symmetry. This corresponds to the nuclear physics case in which the singlet, a0, and triplet, a1, n − p scattering lengths are large with respect to the range of the nuclear interaction. The ratio of the two is about a0/a1 ≈ −4.31. This value defines a plane in which a0 and a1 can be varied up to the unitary limit, 1/a0 = 0 and 1/a1 = 0, maintaining its ratio fixed. Using a spin dependant potential model we estimate the three-nucleon binding energy along that plane. This analysis can be considered an extension of the Efimov plot for three bosons to the case of three 1/2-spin-isospin fermions.\n