1. Spin polarization phenomena in dense nuclear matter Alexander Isayev Kharkov Institute of Physics and Technology Ukraine

2. DSPIN-07 2 OVERVIEW AND MOTIVATION The spontaneous appearance of spin polarized states in nuclear matter is the topic of a great current interest. The spontaneous appearance of spin polarized states in nuclear matter is the topic of a great current interest.  Could nuclear matter, formed in heavy ion collisions at high and intermediate energies, undergo a phase transition to a spin polarized state?

3. DSPIN-07 3 OVERVIEW AND MOTIVATION The spontaneous appearance of spin polarized states in nuclear matter is the topic of a great current interest. The spontaneous appearance of spin polarized states in nuclear matter is the topic of a great current interest.  Could nuclear matter, formed in heavy ion collisions at high and intermediate energies, undergo a phase transition to a spin polarized state?  Spin correlations in the medium significantly influence the neutrino cross section and neutrino luminocity. Hence, different scenarios of supernova explosion and cooling of neutron stars can be realized, depending on whether nuclear matter is spin polarized or not.

4. DSPIN-07 4 OVERVIEW AND MOTIVATION Precisely: Nucleons possess spin. Hence, nuclear matter is a paramagnetic medium and under lowering temperature or changing density nuclear matter can undergo a phase transition to the state with nonequal numbers of spin-up and spin-down nucleons: spin polarized state. Usually: majority of neutron spins are aligned in the same direction as majority of proton spins (like ferromagnetic ordering). Precisely: Nucleons possess spin. Hence, nuclear matter is a paramagnetic medium and under lowering temperature or changing density nuclear matter can undergo a phase transition to the state with nonequal numbers of spin-up and spin-down nucleons: spin polarized state. Usually: majority of neutron spins are aligned in the same direction as majority of proton spins (like ferromagnetic ordering). Another possibility: majority of neutron spins and majority of proton spins have the opposite directions (like antiferromagnetic ordering). Main emphasis: the structure of a ground state of nuclear matter, whether it is spin polarized or not and what type of spin ordering can be realized. Main emphasis: the structure of a ground state of nuclear matter, whether it is spin polarized or not and what type of spin ordering can be realized.

5. DSPIN-07 5 OVERVIEW AND MOTIVATION The possibility of ferromagnetic phase transition in nuclear and neutron matter:  Calculations of magnetic susceptibility with Skyrme effective forces A. Viduarre, J. Navarro, and J. Bernabeu, Astron. Astrophys. 135, 361 (1984). Ferromagnetic transition occurs at  The Fermi liquid criterion for the ferromagnetic instability in nuclear matter with Skyrme interaction is reached at being nuclear matter saturation density. being nuclear matter saturation density. S. Reddy, M. Prakash, J.M. Lattimer, and J.A. Pons, Phys. Rev. C 59, 2888 (1999).

6. DSPIN-07 6  In the models with realistic nucleon-nucleon (NN) interaction the ferromagnetic phase transition seems to be suppressed up to densities well above. I. Vidana, A. Polls, and A. Ramos, Phys. Rev. C 65, 035804 (2002). S. Fantoni, A. Sarsa, and E. Schmidt, Phys. Rev. Lett. 87, 181101 (2001).

7. DSPIN-07 7 Here: the study of spin polarizability of nuclear matter with the use of effective NN interaction (Skyrme, Gogny effective forces). Framework for consideration: Fermi liquid (FL) description of nuclear matter, allowing to obtain the self- consistent equations for the order parameters, and, after solving them, to calculate the free energy and to make conclusions concerning thermodynamic stability of different phases.

8. DSPIN-07 8 Basics of Formalism Normal states of nuclear matter are described by the normal distribution function of nucleons The matrix self-consistent equation for in the state of thermodynamic equilibrium The single-particle energy and is the density matrix of the system

9. DSPIN-07 9 Structure of the normal distribution function and single particle energy are the Pauli matrices in spin and isospin spaces Normalization conditions for the distribution functions - the isospin asymmetry parameter FM spin order parameter AFM spin order parameter In symmetric nuclear matter with FM ordering: AFM ordering:

10. DSPIN-07 10 The energy functional of the system: FL amplitudes describe density, spin, isospin and spin-isospin correlations in nuclear medium.

11. Self-consistent equations Set of integral equations to be solved self-consistently. Numerical procedure: iterations on the Gaussian grid in momentum space until convergence with required accuracy is achieved.

12. DSPIN-07 12 Amplitude of NN interaction for Skyrme effective force Amplitude of NN interaction for Gogny effective force In numerical calculations: SLy4, SkI3, SkI5 parametrizations Two Gaussian terms reflect the finite range character of the Gogny interaction. In numerical calculations: D1S force. Neutron and proton spin polarization parameters:

13. DSPIN-07 13 Neutron and proton spin polarization parameters as functions of density at zero temperature SLy4: 1. Antiparallel ordering only! 2. Even small admixture of protons to neutron matter strongly decreases the onset density of spin instability. 3. Protons become totally polarized in a narrow density domain at strong isospin asymmetry, unlike to neutrons.

14. DSPIN-07 14 SkI5: 1. Parallel ordering only! 2. Small admixture of protons to neutron matter insignificantly change the onset density of spin instability. Neutron and proton spin polarization parameters as functions of density at zero temperature

15. DSPIN-07 15 Total energy per nucleon, measured from its value in the normal state, as a function of density at T=0 The energy gain per nucleon is decreased with isospin asymmetry for SLy4 force while it is increased for SkI5 force.

16. DSPIN-07 16 SkI3: 1. Transient behavior from antiparallel ordering to parallel ordering under increasing density. 2. There are no long tails in the density profile of neutrons at strong isospin asymmetry. 3. The energy gain is increasing function of isospin asymmetry at the given density.

17. DSPIN-07 17 AFM spin polarization parameter as a function of density at zero temperature for D1S Gogny force and SkM*, SGII Skyrme forces. For D1S force only AFM spin ordering is realized and there are no solutions, corresponding to FM spin ordering. Symmetric nuclear matter

18. DSPIN-07 18 Total energy per nucleon, measured from its value in the normal state, for the AFM spin state as a function of density at zero temperature for D1S Gogny force and SkM*, SGII Skyrme forces.

19. CONCLUSIONS Thus, modern effective nuclear forces being relevant for calculations at wide range of isospin asymmetries and high densities provide us with different possible scenarios of a phase transition to a spin polarized state in dense nuclear matter: Thus, modern effective nuclear forces being relevant for calculations at wide range of isospin asymmetries and high densities provide us with different possible scenarios of a phase transition to a spin polarized state in dense nuclear matter: a) nuclear matter with SLy4 Skyrme and D1S Gogny interactions undergoes at some critical density a phase transition to a spin polarized state with the oppositely directed spins of neutrons and protons; b) for SkI5 interaction, a spin polarized state with the like-directed neutron and proton spins is formed; c) nuclear matter with SkI3 interaction 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.

20. DSPIN-07 20 The different behavior at high densities of the interaction amplitudes, describing spin-spin and spin-isospin correlations, lays behind this divergence in calculations with different effective potentials. These results clearly indicate the necessity to construct a new generation of the energy functionals with the properly constrained time-odd part. Probably, these constraints will be obtained from the data on the time decay of magnetic field of isolated neutron stars. S.B. Popov and M.E. Prokhorov, Surveys in High Energy Physics 15, (2001) 381.