Spontaneous magnetization of quark matter in inhomogeneous chiral phase

Spontaneous magnetization of quark matter in

inhomogeneous chiral phase

R. YoshiikeCollaborator: K. Nishiyama, T. Tatsumi

(Kyoto University)

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 phase0

Current quark mass

SSB Broken phase

0

Constituent quark mass

chiral phase transition line

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)

35352 iiGiLMF

Dual chiral density wave(DCDW) condensate

order parameters: Δ, q

cf. conventional broken phase: .const r

35iz

[G. Basar, et al. (2009)]

Inhomogeneous chiral condensate

35ii r embed the solution of 1+1 dimension

iqzeii 35r

Inhomogeneous chiral phase in QCD phase diagram

Inhomogeneous chiral phase can exist in neutron stars!

T

μ

DCDW phase

Lifshitz point

3.6ρ0 ～ 5.3ρ0

Tri-critical point

restored phase

broken phase

T

μ

2nd

1st

[E. Nakano, T. Tatsumi (2005)]

several ρ0 ～ 10ρ0

・ Homogeneous chiral phase (conventional broken phase) ・・・ Δ≠0, q=0・ DCDW phase ・・・ Δ≠0, q≠0・ Restored phase ・・・ Δ=0 iqze r

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 ～ 1012G

(magnetars ～ 1015G)

However, the origin of the magnetic field hasn’t been unraveled.

Phase structure of quark matter in the magnetic fieldThe systems where quark matter can exist in the magnetic field

Neutron stars, Heavy ion collision, Early universe, etc…

T

μB ？

Relevant problem…

Motivation and goals

Thermodynamic potential in the magnetic field

Lagrangian

G

mqziqzmDiLDCDW 4

sincos2

35 )2( Gm

35i

[I. E. Frolov, et al. (2010)]Landau level

1,1,, pnE

2

22

22

222

qpm

nBeq

pm f

(n=1,2, ・・・ ) 　・・・ symmetry

(lowest Landau level(LLL), n=0) 　・・・ asymmetry

0,,0 BxALandau gauge:

E

0

LLL

asymmetric about zero

2qm

2qm

Thermodynamic potential in the magnetic field

Anomaly by the spectral asymmetryAnomalous baryon number

nomNN

cf. [A. J. Niemi, G. W. Semenoff (1986)]

λk ・・・ eigenvalue of Hamiltonian

In this case

2

1,,

0 2sign

22sign lim

eBq

EEdpeB sLLL

pLLLp

skk

[T. Tatsumi, et al. (2014)]

Regularization on the energy

cf. chiral Lagrangian [D. T. Son, M. A. Stephanov,(2008)]

Thermodynamic potential

2210 ,;,,;,,;,,;,, BemqTBemqTmqTmqBT ff Spectral asymmetry of LLL

Regularizing on the energy, it becomes physically correct.

0 mq-independent

Spontaneous magnetizationStationary condition 2210 ,,,,, eBTmeBTmTmBTm

2210 ,,,,, eBTqeBTqTqBTq

001

0

,;,,,

qmTeB

BTM

B

T=0

m(0)

q(0) M

μ(MeV)μ(MeV)

QM has the spontaneous magnetization in DCDW phase! ～ 1017G

0

,

,;,,

mq

mqBT

2210 eBeB

(MeV) (MeV2)

～ (m(0))2

Generalized Ginzburg-Landau expansion

Thermodynamic potential around the Lifshitz point

q

m restored

4

2

homo.

DCDW

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)

(B=0)

　　 LP(α2=α4=0)



q

m restored

4

2

homo.

DCDW

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)

(B=0)

　　 LP(α2=α4=0)



q

m restored

4

2

homo.

DCDW

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)

(B=0)

　　 LP(α2=α4=0)



q

m restored

4

2

homo.

DCDW

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)

(B=0)

　　 LP(α2=α4=0)



q

m restored

4

2

homo.

DCDW

① 24

62 8

3

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)

2nd order phase transition

Phase boundaries

(B=0)

　　 LP(α2=α4=0)

①

①



q

m restored

4

2

homo.

DCDW

① 24

62 8

3

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0

　　 LP(α2=α4=0)

(MeV2) (MeV)


②24

62

...1717.0

1st order phase transition

Phase boundaries

(B=0)

　　 LP(α2=α4=0)

②

②



q

m restored

4

2

homo.

DCDW

① 24

62 8

3

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0 (MeV2) (MeV)


②24

62

...1717.0

1st order phase transition

③ 02

(conventional) 2nd order phase transition

Phase boundaries

(B=0)

③

③



q

m restored

4

2

homo.

DCDW

4224610

26

4325

2248

24

23

26

220

2

13

9

10

6

13

9

5

4

1

27

5

2

1

qmqmmeBqmqmeB

qmmeBqmeBmeB

eB=0 (MeV2) (MeV) eB=60(MeV)2

～ 1015

Ｇ

homo.→ ＤＣＤＷ

m

q

　　 LP(α2=α4=0)

　　 LP(α2=α4=0)

μ-T plane mappingLP(α2=0,α4=0) 　→　 LP( 　　　　　　　　　　　　　　 )T

μ

m

q

eB=0

m

Switching on B, DCDW region expands andhomogeneous phase changes to DCDW phase!

q

homo.→DCDW

MeVTMeV 130,480

eB=60(MeV)2 ～ 1015 Ｇ

(MeV)

MeVT 125

m

q

1st 2ndμ(MeV)

(MeV) 0B

Magnetic properties around Lifshitz point

M

χ

μ(MeV)

0

2

2

B

eB

Magnetic susceptibility does not diverge but has discontinuity

MeVT 125 χ M(MeV2)

0

BeB

M

Spontaneous magnetization Magnetic susceptibility

T

μ

T=125MeV

Ferromagnetic transition point

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

Documents

Spontaneous magnetization of quark matter in inhomogeneous chiral phase