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**R. Yoshiike Collaborator: K. Nishiyama, T. Tatsumi (Kyoto University)**

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

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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) Inhomogeneous chiral condensate embed the solution of 1+1 dimension cf. conventional broken phase: Dual chiral density wave(DCDW) condensate z order parameters: Δ, q [G. Basar, et al. (2009)]

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**Inhomogeneous chiral phase in QCD phase diagram**

・Homogeneous chiral phase (conventional broken phase) ・・・Δ≠0, q=0 ・DCDW phase・・・Δ≠0, q≠0 ・Restored phase・・・Δ=0 T μ DCDW phase Lifshitz point 3.6ρ0～5.3ρ0 Tri-critical point restored phase broken phase 2nd 1st [E. Nakano, T. Tatsumi (2005)] Inhomogeneous chiral phase can exist in neutron stars! several ρ0～10ρ0

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**Strong magnetic field in neutron stars**

Motivation and goals Strong magnetic field in neutron stars Surface of neutron stars ～1012G (magnetars ～1015G) However, the origin of the magnetic field hasn’t been unraveled. Goals ・investigate the magnetic properties of quark matter in DCDW phase ・explain the origin of strong magnetic field in neutron stars Relevant problem… 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 ？

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**Thermodynamic potential in the magnetic field**

Lagrangian Landau gauge: [I. E. Frolov, et al. (2010)] Landau level (n=1,2,・・・) ・・・symmetry (lowest Landau level(LLL), n=0) ・・・asymmetry LLL E 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)] λk・・・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 (MeV) (MeV2) T=0 m(0) q(0) M μ(MeV) ～(m(0))2 QM has the spontaneous magnetization in DCDW phase! ～1017G

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) m restored LP (α2=α4=0) DCDW homo. q LP (α2=α4=0)

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) m restored LP (α2=α4=0) DCDW homo. q LP (α2=α4=0)

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) m restored LP (α2=α4=0) DCDW homo. q LP (α2=α4=0)

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) m restored LP (α2=α4=0) DCDW homo. q LP (α2=α4=0)

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) Phase boundaries m restored ① ① LP (α2=α4=0) DCDW 2nd order phase transition homo. q ① LP (α2=α4=0)

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) Phase boundaries m restored ① LP (α2=α4=0) DCDW 2nd order phase transition ② homo. ② 1st order phase transition q LP (α2=α4=0) ②

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point (B=0) eB=0 (MeV2) (MeV) Phase boundaries m restored ① DCDW ③ 2nd order phase transition homo. ② 1st order phase transition q ③ ③ (conventional) 2nd order phase transition

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**Generalized Ginzburg-Landau expansion**

Thermodynamic potential around the Lifshitz point eB=0 (MeV2) (MeV) eB=60(MeV)2 ～1015Ｇ m restored m LP (α2=α4=0) DCDW homo. q q LP (α2=α4=0) homo.→ＤＣＤＷ

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**μ-T plane mapping Switching on B, DCDW region expands and**

LP(α2=0,α4=0) → LP( ) (MeV) m m m q 1st 2nd μ(MeV) (MeV) eB=0 eB=60(MeV)2～1015Ｇ q q homo.→DCDW Switching on B, DCDW region expands and homogeneous phase changes to DCDW phase!

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**Magnetic properties around Lifshitz point**

Spontaneous magnetization Magnetic susceptibility χ M(MeV2) χ T μ T=125MeV M μ(MeV) Ferromagnetic transition point Magnetic susceptibility does not diverge but has discontinuity

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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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