1. Ferromagnetism in quark matter and origin of magnetic field in compact stars Toshitaka Tatsumi (Kyoto U.) (for a recent review, hep-ph/0506002) I. Introduction Magnetic phase diagram of QCD in T-  plane. Pion condensation Ferromagnetism Spin density wave RHIC and Compact stars Discovery of magnetars CSC QCD at finite density and/or temperature Chiral symmetry restoration New class of compact stars with high B

2. A naïve working hypothesis, conservation of magnetic flux during stellar evolution, looks to break down for magnetars. Three groups of compact stars Microscopic origin of the magnetic field Long standing problem for the origin of B

3. II. Relativistic ferromagnetism in quark matter ref. T.T., PLB489(2000) 280. T. Maruyama and T.T., NPA 693 (2001) 710. Perturbative treatment in analogy with Bloch’s mechanism (Pauli principle) Fock exchange int. favors spin alignment for the Coulomb (gauge) interaction. Notes (QCD): Kinetic energy increases, while the Weakly first order meta-stable OGE “Spin” is not a good quantum number

4. Then Deformation of the Fermi seasSelf-consistent (H-F) approach (Axial-vector mean-field)

5. An effective model with zero-range approx. for OGE (contact interaction) (i) Is the genuine Fermi sea effect (ii)Vacuum pol. works against alignment. (iii) Mass effect: Critical density Careful treatment of the interaction range! Second order Self-consistent eq. FM Qualitative change may happen. (spin mean-value over both Fermi seas 、 and Dirac sea)

6. III. Color magnetic superconductivity ref. E. Nakano, T. Maruyama and T.T., PRD68 (2003) 105001. Coexistence of FM and CSC OGE interaction (Nambu-Gorkov) FM CSC One may also expect CSC in high-density quark matter.

7. Fock self-energy FM: CSC:

8. Diquark pairing structure

9. IV. Chiral-symmetry restoration and Spin density wave ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006. Chiral manifold Restoration paths Density-wave instability before chiral-symmetry restoration or another restoration path due to pseudo-scalar density

10. NJL model for simplicity. MFA and the Weinberg transformation, gives Axial-vector MF

11. Spin density wave (c.f. Overhauser) DCDW

12. V. Summary and Concluding remarks A magnetic phase diagram of QCD Ferromagnetism (FM) and Spin density wave (DCDW=SDW) Resemblance of DCDW to pion condensation (PIC) in the hadron phase Hadron-quark continuity? CSC We have discussed the magnetic aspect of quark matter. Magnetized strangelets (small “magnets”) Relativistic heavy-ion collisions Primordial magnetic field as a relic of hadron-quark phase transition in the early universe. CSC Implications:

13. Hierarchy of magnetic field observed in compact stars NG excitations (spin wave, phason) in the magnetic phase. Magnetic domain formation recycled pulsars radio pulsars magnetars Magnetic field: Period: Age: As a direct implication Magnetars=Quark stars Field decay?