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Topological aspects of the Inhomogeneous chiral phases

Topological aspects of the Inhomogeneous chiral phases and implication on compact stars. T. Tatsumi (Kyoto U.). contents: I Introduction II Inhomogeneous chiral phase in the magnetic field III Spectral asymmetry in the inhomogeneous chiral phase

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Topological aspects of the Inhomogeneous chiral phases

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  1. Topological aspects of the Inhomogeneous chiral phases and implication on compact stars T. Tatsumi (Kyoto U.) contents: I Introduction II Inhomogeneous chiral phase in the magnetic field III Spectral asymmetry in the inhomogeneous chiral phase IV Novel tricritical (Lifshitz) point in the presence of the magnetic field V Phenomenological implications VI Summary and concluding remarks In collaboration with R. Yoshiike, K. Nishiyama, S. Karasawa and T. Muto 1

  2. I. Introduction QCD phase diagram (For a review , K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74, 014001 (2011)) Heavy-ion collision Neutron Stars G. Endroedi al., arXiv:1311.0648. A.Bazabov et al., PRD 80(2009) 014504. QCD vacuum in the magnetic field 2

  3. Chiral phase transition CCP FM? Generalized chiral order-parameter: Chiral transition There appear inhomogeneous chiral phases in the vicinity of the chiral transition : Compact stars (B. Ruester) Critical point (CCP) as Lifshitz point (LP) H QCD phase diagram Two typical forms of the condensates have been studied within NJL model: (T. T. and E. Nakano, hep-ph/0408294. E. Nakano and T. T., PRD 71 (2005) 114006.) Dual chiral density wave (DCDW): Real kink crystal (RKC): DCDW (D.Nickel, PRL 103(2009) 072301; PRD 80(2009) 074025.) For a recent review, M. Buballa and S. Carignano, arXiv:1406.1367 3

  4. DCDW(dual chiral density wave) T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006. SSB z Usual chiral transition　（T=0) DCDW Instability of the Fermi surface due to the ph excitations (Nesting effect) It provides another path of the restoration of chiral symmetry 4

  5. Results within 2 flavor NJL model Tricritical point=Lifshitz point Chiral limit restored SSB Critical end point (CEP) DCDW DCDW m S. Karasawa and T.T., arXiv:1307.6448. ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006. 5

  6. Inhomogeneous (spatially dependent) order parameter in the vicinity of the phase transition Common feature in condensed matter physics ?! FFLO state in superconductor Spin density wave (SDW) from ferromagnetic phase General solutions of the one dimensional order Chiral symmetry (SSB) Twisted kink crystal condensate G.Basar and G.V.Dunne, PRL 100(2008)2004004; PRD 78(2008) 065022. z 6

  7. II Inhomogeneous chiral phase in the magnetic field S. P. Klevansky and R.H. Lemmer, PRD 39 (1989) 3478. H. Suganuma and T.T., Ann.Phys. 208 (1991) 371. B V.P. Gusynin, V.A. Milansky, I.A. Shovkovy, NPB 462 (1996) 249 E Cooper pair S.P. Klevansky, RMP 64(1992) 649. Phase transition ・Enhancement of SSB or Magnetic catalysis 7

  8. 1/2 Dirac operator in the magnetic field: (eH) Using the Landau gauge, T=0 Dirac Hamiltonian with the inhomogeneous condensate: DCDW （DCDW） Hybrid condensate restored RKC m Competition between DCDW and RKC Nishiyama’s talk in Session 6 H=0 I.E. Frolov et al, PRD 82, 076002 (2010) 8

  9. III Spectral asymmetry in the inhomogeneous chiral phase ex) In the case of DCDW [I.E. Frolov et al., Phys. Rev. D82 (2010) 076002] Energy spectrum : Cf: Landau levels: Spectral asymmetry DCDW m Spectrum is symmetric about 0 0 m q/2 LLL 9

  10. It is known that spectral asymmetry induces the anomalous baryon number. e.g. Chiral bag (Goldstone & Jaffe, PRL 51(1983) 1518.) MIT bag Skyrmion (“Baryon number of the vacuum”) Atiyah-Patodi-Singer h invariant (Atiyah et al, Proc.CambridgePhils. Soc. 77(1975) 42;78(1975) 405;79 (1976) 71): Baryon number in the thermodynamic limit: (A.J. Niemi and G.W. Semenoff, Phys. Reports 135 (1986) 99.) Topological one which must be consistent with the thermodynamic relation, 10

  11. Direct evaluation of hHfor DCDW (LLL only) E+=q/2+m Using the Mellin transform q/2 m E-=q/2-m ０ LLL his independent of m. Thus topological baryon-number density can be written as 11

  12. Important implications of the spectral asymmetry ・It is closely related to chiral anomaly ・It induces the anomalous quark number, and broadens the inhomogeneous phase Nishiyama’s talk in Session 6 ・It gives a new tricritical (Lifshitz) point on the QCD phase diagram ・It provides us with a possibility of the spontaneous magnetization Yoshiike’s talk Anomaly in quark matter D.T. Son and A.R. Zhitnitsky, PRD 70, 074018 (2004). D.T. Son and M.A. Stephanov, PRD 77, 014021 (2008) It well-known that (chiral) anomaly plays an important role in hadron mass spectrum, hadron dynamics such as or hadron models such as skyrmion. However its role in quark matter or many-body dynamics has not been well-understood yet. 12

  13. Chiral anomaly and DCDW (T.T.,K. Nishiyama and S. Karasawa, EPJ Web of Cof. 71, 00131 (2014); arXiv:1405.2155) For the DCDW-type configuration (D.T. Son and M.A. Stephanov,PRD 77(2008) 014021.) Spontaneous Magnetization? Yoshiike’s talk 13

  14. IV Novel tricritical point in the presence of the magnetic field Generalized Ginzburg-Landau (gGL ) theory: The coefficients are evaluated in terms of the Green function in the presence of the magnetic field [Schwinger, (1951)], (sum over the Landau levels) (A. Chodos et al (1990)) 14

  15. The “odd” terms survive in the presence of the magnetic field, due to the spectral asymmetry of the LLL levels. So, we can define the new tricritical point (Lifshitz point) by The odd terms arise as a consequence of the asymmetric appearance of M,M* in the Hamiltonian of the each sector of the Landau levels. If spectrum is symmetric, the “odd” terms vanish: Then the Lifshitz point is given by the conditions: which meets the tricritical point of the chiral transition in the chiral limit [Nickel, (2009)]. Relation between “odd” terms and chiral anomaly? 15

  16. E+=q/2+m m Near the Lifshitz point, m,qare small and we can assume without loss of generality. q/2 E-=q/2-m ０ topological non-topological LLL to to Using the thermodynamic relation, we can see the spectral asymmetry of LLL gives “odd” terms; anomaly effect is already included in the “odd” terms. 16

  17. LLL contribution to (no divergence) This may be natural to consider that DCDW or pseudoscalar condensate must vanish at m=0 due to parity symmetry, which means the spectral asymmetry also vanishes there. 0.05 T (tri-gamma function) 0.1 (higher Landau levels don’t contribute to a3） 0.15 m 17

  18. which includes contributions from higher Landau levels as well as LLL. Lifshitz line Phase diagram in the m-T-Hspace Lifshitz point Usual chiral transition 18

  19. V Phenomenological implications (i) Spontaneous magnetization Magnetars Strong magnetic field in compact stars One of the long standing but very important problems in physics Since the first discovery of pulsar! Origin: • Fossil field • Dynamo scenario (crust) • Microscopic origin (core) cf T.T., PLB 480, 280 (2000) Spontaneous magnetization in DCDW? 19

  20. (ii) Cooling of hybrid stars CAS A 3C58 Vela Cas A Suggest nucleon superfluidity? 20

  21. Rapid cooling mechanism (TT and T. Muto, PRD89, 103005 (2014) .) Usually neutrino emission through the quark beta decay is strongly prohibited in neutron or hybrid stars, while it may give a large emission rate. DCDW catalyzed cooling e The DCDW state can be represented as a chirally rotated state u d DCDW “pole” contribution ,which means DCDW modifies the momentum conservation at the weak-interaction vertex. (O.Maxwell et al(MBCDM), Ap.J.216(1977)77 cf pion cooling T. Muto and TT, PTP 80(1988)28.) 21

  22. General formula is very complicated, but we can easily estimate the emissivities near the phase boudaries: Near the onset density Near the termination density ・Emissivitiesare large in both cases, comparable with quark Durca or pion cooling ・This mechanism works in the limited density region Cooling of neutron stars 22

  23. VI Summary and concluding remarks: ・We have studied dual chiral density wave (DCDW) in the presence of the magnetic field. ・Spectral asymmetry arises in the LLL and anomalous baryon number is induced. DCDW phase is greatly extended due to the magnetic field. T ・Generalized Ginzburg-Landau analysis suggests, that there emerges a novel tricritical point　（line）on the H-T plane (m=0). It should be interesting to explore it in the lattice QCD simulations, free from the sign problem. m DCDW Realistic study is needed to include the current mass or mass dependence. H 23

  24. ・Stability of the one dimensional order Quasi long range order (QLRO) Landau-Peierls theorem Magnetic field Long range order due to the spectral asymmetry of LLL ・ Phenomenological implications (i) Spontaneous magnetization is possible? origin of magnetic field in compact stars (ii) Novel cooling mechanism works there and emissivity is almost the same as the pion cooling or quark cooling Change of the dispersion relation of the NG mode anisotropic but quadratic in momentum! 24

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