1. 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)

2. 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

3. 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)]

4. 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

5. 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 ？

6. 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

7. 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

8. 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

9. 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)

10. 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)

11. 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)

12. 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)

13. 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)

14. 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

15. 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

16. 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

17. 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!

18. 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

19. 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