1. Magnetic aspects of QCD and compact stars T. Tatsumi Department of Physics, Kyoto University I.Introduction and motivation II.Chiral symmetry and spin density wave (SDW) III.Ferromagnetism (FM) and magnetic susceptibility IV.Screening effects for gluons V.Magnetic properties at T=0 VI.Finite temperature effects and Non-Fermi-liquid behavior VII.Summary and concluding remarks T.T., Proc. of EXOCT07 (arXiv:0711.3348) T.T. and K. Sato., Phys. Lett., B663 (2008) 322. K. Sato and T.T., Prog. Theor. Phys. Suppl. 174 (2008) 177. T.T. and K. Sato, Phys. Lett. B672(2009) 132. K. Sato and T.T., Nucl. Phys. A826 (2009) 74. T.T., Proc. of CSQCDII (2010) in press.

2. Meson (Pion) condensation (PIC) Antiferromagnetism (AFM) Deconfinement Ferromagnetism (FM) Chiral restoration Spin density wave (SDW) Color superconductivity. CSC Chiral restoration Magnetic phase diagram of QCD Critical end-point I. Introduction and motivation Deconfinement (I) Phase diagram of QCD ? Spin degrees of freedom

3. Origin: (i)Fossil field (ii)Dynamo scenario (crust) (iii)Microscopic origin (core) (II) Strong magnetic field in compact stars Magnetars Its origin is a long-standing problem since the first discovery of pulsars. Recent discovery of magnetars seems to revive this issue

4. Ferromagnetism or spin polarization For recent references, I.Bombaci et al, PLB 632(2006)638 G.H. Bordbar and M. Bigdeli, PRC76 (2007)035803 Spontaneous magnetization of quark matter or ferromagnetism in quark matter Nuclear matter calculations have shown negative results It would be rather natural to attribute its origin to strong interaction

5. Some ideas in QCD E.J. Ferrer, de la Incera, PRL 97(2006) 122301; PRD 76(2007) 045011; PRD 76(2007)114012. arXiv:0810.3886 Bloch mechanism Cf. Other mechanism in CSC: ・ Gluon condensate D.T. Son, M.A. Stephanov, PRD 77(2008)014021. ・ Axial anomaly in CFL X B B

6. A typical example of the condensed phase : Liquid crystal with antiferromagnetic order II. Chiral symmetry and Spin density wave (SDW) T.Takatsuka et al., Prog.Theor.Phys. 59(1978) 1933. T.Suzuki et al., arXiv:nucl-th/9901097

7. ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006. A density-wave instability before/around chiral-symmetry restoration or another restoration path due to pseudo-scalar density A magnetic phase in the intermediate densities Chiral symmetry restoration and SDW Chiral-restoration path SDW

8. Remarks: (i)Nesting (Overhauser, Peiels) is the key mechanism for generating SDW Level crossing of the energy spectrum near the Fermi surface Model indep. SDW,CDW kFkF q A.W. Overhauser, PRL 4(1960) 462. R.E. Peierls, Quntum Theory of Solids (1955)

9. (ii) Similar idea (iii) Similarity to LOFF in superconductor

12. Spin density wave （ SDW ） (c.f. Overhauser) DCDW

13. (iii) Phason and spin-wave as NG modes in SDW G.Gruener, Rev.Mod.Phys. 66(1994) 1. It would be interesting to see that both modes have the linear dispersion relation: Cf. Spin waves in FM and AFM Counting rule of NG bosons

14. T.T. PLB489(2000)280. T.T.,E. Nakano and K. Nawa, Dark matter, p.39 (Nova Sci. Pub., 2005) Fock exchange interaction is responsible to ferromagnetism in quark matter (Bloch mechanism) c.f. Ferromagnetism of itinerant electrons (Bloch,1929) v v vv qq’ q OGE k k q q Is there ferromagnetic instability in QCD? III. Ferromagnetism (FM) and magnetic susceptibility

15. Weakly first order c.f. A.Niegawa, Prog. Theor. Phys. 113(2005)581, which also concludes ferromagnetism at low density, by the use of the resummation technique. Magnetars as quark stars

16. Relativistic Fermi liquid theory (G.Baym and S.A.Chin, NPA262(1976)527.) No direct int. Fock exchange int. Color symmetric int.: No flavor dep. In the following we are concerned with only one flavor. Ferromagnetism in gauge theories

17. 0 spin susceptibility Magnetization Change of the distribution function Dirac magneton Magnetic susceptibility :response to the external magnetic field

18. ０ ０ Free energy which also measures the curvature of the free energy at the origin spontaneous magnetization N(T): effective density of states at the Fermi surface Fermi velocity f: Landau parameters Magnetic (spin) susceptibility in the Fermi liquid theory Infrared (IR) singularities

19. k q k q  p  p=k-q (Debye mass) (Landau damping) Gauge choice （ i) Debye screening in the longitudinal (electric) gluons improve IR behavior (ii ) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence Non-Fermi-liquid behavior HDL resummation IV Screening effects

20. V. Magnetic properties at T=0 Quasiparticle interaction: longitudinal transverse Susceptibility Screening effect log div Simple OGE cancellation k q k q  p 

21. s quark only u,d,s symmetric matter Ferromagnetic phase Paramagnetic phase without screening suppressionenhancement ● ● N F dependence

22. Ferromagnetism Paramagnetism Large fluctuations (or paramagnon) cf. emissivity ( talk by S. Reddy) SDW? cf in electron gas FM(Bloch) SDW(Overhauser) low density high-density Note that this SDW has nothing to do with chiral symmetry, but nesting is also important

23. （ i) Debye screening in the longitudinal (electric) gluons improve IR behavior (ii ) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence (iv) Results are independent of the gauge choice  (iii) Divergences cancel each other to give a finite  Non-Fermi-liquid behavior Screening effect Some features: To summarize:

24. VI. Finite temperature effects and Non-Fermi-liquid behavior ・ We consider the low T case, T/  but the usual low-T expansion cannot be applied. ○ Density of state: ・ Quasiparticle energy exhibits an anomalous behavior near the Fermi surface ○

25. Quark self-energy Schwinger-Dyson Anomalous term (C. Manuel, PRD 62(2000) 076009) One loop result:

26. Role of transverse gluons =relevant interactions in RG Non-Fermi liquid behavior ・ Specific heat ・ Gap equation How about susceptibility? Peculiar temperature dep. of susceptibility Curie temperature (D.T. Son, PRD 59(1999)094019) (A. Ipp et al., PRD 69(2004)011901) Effective coupling isInfrared free (Schaefer, K. Schwenzer, PRD 70(2004) 054007)

27. T-indep. term T 2 -term Non Fermi-liquid effect Magnetic susceptibility at T>0 Cf. paramagnon effect

28. Paramag. FM Magnetic phase diagram of QCD Curie (critical) temperature should be order of several tens of MeV. Non-Fermi-liquid effect

29. VII. Summary and concluding remarks ・ We have considered magnetic susceptibility  q=0) of QCD within Fermi-liquid theory Roles of static and dynamic screening are figured out: Static Dynamic Novel non-Fermi liquid effect! ・ Since the order parameter is color singlet, FM and SDW survive even in the large Nc limit. ・ Observational signatures of magnetic phases Thermal evolution as well as magnetic evolution Novel mechanism of neutrino emissivity!? Specific heat and thermal conductivity? ・ Spin wave or magnons in FM ・ SDW and phasons （ T.T., arxiv:07113349 ） ・ Possibilities and properties of SDW and FM have been discussed