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Spontaneous magnetization of quark matter in inhomogeneous chiral phase R. Yoshiike Collaborator: K. Nishiyama, T. Tatsumi (Kyoto University)

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QCD phase diagram and chiral symmetry Various phase structure of quark matter [K. Fukushima, T. Hatsuda (2011)] ・ Hadronic phase ・ Quark-gluon plasma ・ Color superconductor etc… Chiral symmetry Restored phase Current quark mass SSB Broken phase Constituent quark mass chiral phase transition line

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What’s inhomogeneous chiral phase? “new phase in the high density region of the QCD phase diagram” NJL model in mean field approximation(2-flavor) Dual chiral density wave(DCDW) condensate order parameters: Δ, q cf. conventional broken phase: z [G. Basar, et al. (2009)] Inhomogeneous chiral condensate embed the solution of 1+1 dimension

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Inhomogeneous chiral phase in QCD phase diagram Inhomogeneous chiral phase can exist in neutron stars! T μ DCDW phase Lifshitz point 3.6ρ 0 ～ 5.3ρ 0 Tri-critical point restored phase broken phase T μ 2nd 1st [E. Nakano, T. Tatsumi (2005)] several ρ 0 ～ 10ρ 0 ・ Homogeneous chiral phase (conventional broken phase) ・・・ Δ≠0, q=0 ・ DCDW phase ・・・ Δ≠0, q≠0 ・ Restored phase ・・・ Δ=0

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Strong magnetic field in neutron stars Goals ・ investigate the magnetic properties of quark matter in DCDW phase ・ explain the origin of strong magnetic field in neutron stars Surface of neutron stars ～ G (magnetars ～ G) However, the origin of the magnetic field hasn’t been unraveled. Phase structure of quark matter in the magnetic field The systems where quark matter can exist in the magnetic field Neutron stars, Heavy ion collision, Early universe, etc… T μ B ？ Relevant problem… Motivation and goals

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Thermodynamic potential in the magnetic field Lagrangian [I. E. Frolov, et al. (2010)] Landau level (n=1,2, ・・・ ) ・・・ symmetry (lowest Landau level(LLL), n=0) ・・・ asymmetry Landau gauge: E 0 LLL asymmetric about zero

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Thermodynamic potential in the magnetic field Anomaly by the spectral asymmetry Anomalous baryon number cf. [A. J. Niemi, G. W. Semenoff (1986)] λ k ・・・ eigenvalue of Hamiltonian In this case [T. Tatsumi, et al. (2014)] Regularization on the energy cf. chiral Lagrangian [D. T. Son, M. A. Stephanov,(2008)] Thermodynamic potential Spectral asymmetry of LLL Regularizing on the energy, it becomes physically correct. q-independent

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Spontaneous magnetization Stationary condition T=0 m (0) q (0) M μ(MeV) QM has the spontaneous magnetization in DCDW phase! ～ G (MeV) (MeV 2 ) ～ (m (0) ) 2

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) (B=0) LP (α 2 =α 4 =0)

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) (B=0) LP (α 2 =α 4 =0)

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) (B=0) LP (α 2 =α 4 =0)

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) (B=0) LP (α 2 =α 4 =0)

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW ① eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) 2nd order phase transition Phase boundaries (B=0) LP (α 2 =α 4 =0) ① ①

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW ① eB=0 LP (α 2 =α 4 =0) (MeV 2 ) (MeV) 2nd order phase transition ② 1st order phase transition Phase boundaries (B=0) LP (α 2 =α 4 =0) ② ②

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW ① eB=0 (MeV 2 ) (MeV) 2nd order phase transition ② 1st order phase transition ③ (conventional) 2nd order phase transition Phase boundaries (B=0) ③ ③

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Generalized Ginzburg-Landau expansion Thermodynamic potential around the Lifshitz point q m restored homo. DCDW eB=0 (MeV 2 ) (MeV) eB=60(MeV) 2 ～ Ｇ homo.→ ＤＣＤＷ m q LP (α 2 =α 4 =0) LP (α 2 =α 4 =0)

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μ-T plane mapping LP(α 2 =0,α 4 =0) → LP( ) T μ m q eB=0 m Switching on B, DCDW region expands and homogeneous phase changes to DCDW phase! q homo.→DCDW eB=60(MeV) 2 ～ Ｇ (MeV) m q 1st 2nd μ(MeV) (MeV)

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Magnetic properties around Lifshitz point M χ μ(MeV) Magnetic susceptibility does not diverge but has discontinuity χ M(MeV 2 ) Spontaneous magnetization Magnetic susceptibility T μ T=125MeV Ferromagnetic transition point

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Summary Quark matter in the original DCDW phase has the spontaneous magnetization because of spectral asymmetry. Magnetic susceptibility has discontinuity on the phase transition point. Magnetic field spreads DCDW phase and changes homogeneous phase to DCDW phase. Future work We want self-consistent conclusion taken account for magnetic field by the spontaneous magnetization. Neutron stars

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