Ferromagnetic properties of quark matter

a microscopic origin of magnetic field in compact stars

T. Tatsumi Department of Physics, Kyoto University

I. Introduction and motivationII. Relativistic Fermi-liquid theory and magnetic susceptibilityIII. Screening effects for gluonsIV. Magnetic properties at T=0V. Finite temperature effects and Non-Fermi-liquid behaviorVI. Summary and concluding remarks

T.T., Proc. of EXOCT07 (arXiv:0711.3348)T.T. and K. Sato., Phys. Lett., B663 (2008) 322. K. Sato and T.T. , Prog. Theor. Phys. Suppl. 174 (2008) 177.

T.T. and K. Sato, Phys. Lett. B672(2009) 132.K. Sato and T.T., Nucl. Phys. A (2009) in press.（ arXiv: 0812.1347 ）

30 th anniversary of March 5, 1979 Gamma-ray burst event

Origin:(i) Fossil field(ii) Dynamo scenario (crust)(iii)Microscopic origin (core)

(II) Strong magnetic field in compact stars

I. Introduction and motivation

(I) Phase diagram of QCD

In density-temperature plane Hadron

Quark-Gluon Plasma

T

BFerromagnetism ?

Phase diagram of QCD

Compact stars

~150MeV

CSC

Low temperature and moderate densities Magnetars

No possibility for nuclear matter

For recent references,I.Bombaci et al, PLB 632(2006)638G.H. Bordbar and M. Bigdeli, PRC76 (2007)035803

Spontaneous magnetization of quark matter

Is there ferromagnetic instability in QCD?

It is a long-standing problem since the first discoveryof pulsars.

There have been done many calculations of nuclear matter.

（ ungapped quark matter ）E.Nakano, T. Maruyama, T.T., PRD 68(2003) 105001

Spontaneous spin polarization in quark matter

T.T. PLB489(2000)280.T.T.,E. Nakano and K. Nawa, Dark matter,p.39 (Nova Sci. Pub., 2005)

Fock exchange interaction is responsible to ferromagnetism in quark matter (Bloch mechanism)c.f. Ferromagnetism of itinerant electrons (Bloch,1929)

vv

v v

q q’

q’ q

OGE

k

k

q

q

Ferromagnetism in gauge theories

1/ 2 1/ 2 0c abab baN

(1 ) / 2n p n

Weakly first order

c.f. A.Niegawa, Prog. Theor. Phys. 113(2005)581, which also concludes ferromagnetism at low density,by the use of the resummation technique.

ferro para

Magnetars as quark stars

15 17(10 )O G

(Pauli-Lubanski vector)

II. Relativistic Fermi liquid theory:

Quarks as Quasi-particles:

(G.Baym and S.A.Chin, NPA262(1976)527.)

No direct int. Fock exchange int. (tr 0)

: , ':2 '' ,,

,

1= q q

a ba b

s

c k q

m mf

N E EMf kζ q kζk ζζ q qζζ

Color symmetric int.:

; , ' ; , 'ai bj ij a bf fkζ qζ kζ qζNo flavor dep.

In the following we are concerned with only one flavor.

; , '

( ; ) / ( ; )

( ; ) / ( '; )ai bj

ai E n ai

f ai n bj

kζ qζ

kζ kζ

kζ qζ

4 QED 4i

4

nt

:= / 2Gordon identit

d

d .

y

q

qxL e d

x i

x

A B r

γ

Σ Br

A

:q

2 2/ 2C F FN N Vk k

0L

Fk

0q p -k

Fk

Fk

BFk

Fk

0

spin susceptibilityMagnetization

'','

3

3

)2(

qkk

kkk

kkk

nfqd

N

Bgn

nn

qC

qD

1

exp 1/ 2 1

with the gyromagnetic ratio .

k D q

D

n g B

g

k

2D q

C

gM q q N n n

k k

k

Change of the distribution function

Dirac magneton

Magnetic susceptibility :response to the external magnetic field

, , 0

M

N T B

M

B

M

MF01

０ M

MF01

０

Free energy

which also measures the curvature of the free energy at the origin

-1 or 0

spontaneous magnetization

2

2

1

2

( ) / 1 ( )2

1 = /

2 3s

D q oM

D q

F F

a

C

gN T N T f

g

N kf f

E

N(T): effective density of states at the Fermi surface

sFCFF

k

FC

fkNk

v

kkNTN

F

12

2

2

2

3

,)(

k

, ' 's af f f k q kq kq

Fermi velocity

f: Landau parameters

Magnetic (spin) susceptibility in the Fermi liquid theory

Infrared (IR) singularities

2

screen

4

12( ) (

2 2

,

2

)

,

0

( ) ( ) (

( ) / 2

(

) ,

( )

,

)4

, | |

/

T LT L

T L T

OGE

L

fL D f F

f u d s

FF

DF FT

p pD p P D p P D p

p

D p p

u k

p m g k

pu mEp i

p

k q

kq

p

p=k-q

(Debye mass)

(Landau damping)

Gauge choice

（ i) Debye screening in the longitudinal (electric) gluons

improve IR behavior(ii) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence

Non-Fermi-liquid behavior

HDL resummationIII Screening effects

IV. Magnetic properties at T=0

)()(2

1 002

22

2

2

||||qkqk

Tii

LFC

Ckqk

DMDME

mg

N

Nf

F

Quasiparticle interaction:

1

1 cos kq kqcos1

1

longitudinal transverse

1

Pauli

2

2

2 2

1 (2 )12

1

2l ( 4 2 )

( 0) /

n2

fF

F F

F F

M

C gm E m

E k

E

T

E m m

Susceptibility

Screening effect 4 2( ln )O g g

Pauli : Pauli paramagnetism

2

2

2 F

D

km

log div

Simple OGE

cancellation

k q

kq

p

s quark only

u,d,s 　symmetric matter

Ferromagneticphase

Paramagneticphasewithout

screening

cFk

suppression enhancement

●

●

2ln1

2

2

2 F

D

km

iFiFD Ekg

m ,,flavors

2

22

2

NF dependence

2cFN

（ i) Debye screening in the longitudinal (electric) gluons improve IR behavior

(ii) Transverse (magnetic) gluons only gives the dynamical screening, which leads to IR (Log) divergence

(iv) Results are independent of the gauge choice

(iii) Divergences cancel each other to give a finite

Non-Fermi-liquid behavior

...)ln()()(1 242

spin

Pauli ggOgO

Screening effect

Some features:

To summarize:

V. Finite temperature effects and Non-Fermi-liquid behavior

2 2

1

( )

2 1 ( )

12 ( )

FD q

a al t

Fq

MD

a atl

g N T

N T f f f f

g

N T

3

; ,3

spin dependent FL interaction s.t, :

( )2 / ( )

(2 )

.

s

a al t

a aki c i k k

k

f f

nd kf N f N T

・ We consider the low T case, T/but the usual low-T expansion cannot be applied.

( )sk

3

3

1( ) 2 ,

(2 ) 1 exp( ( ))k

C kk k

nd kN T N n

○ 　 Density of state:

・ Quasiparticle energy exhibits an anomalous behavior near the Fermi surface

○

Quark self-energy

D

SSchwinger-Dyson

2

2reg

2

( ) ln1

For

Re ( ) Re ( ) ( ),

1, /

2 | |

2

f F

cf F F F

c

C g

Nu k E

N

u

C

32

3( ) ( ) ( )

(2 )n

d qp e T S p q D q

Anomalous term

(C. Manuel, PRD 62(2000) 076009)

One loop result:

Role of transverse gluons=relevant interactions in RG

Non-Fermi liquid behavior

・ Specific heat

・ Gap equation

How about susceptibility?

Peculiar temperature dep. of susceptibility

Curie temperature

lnVC T T

2 2exp ( 4)( 1) /16 exp( 3 / 2 )C gN

(D.T. Son, PRD 59(1999)094019)

(A. Ipp et al., PRD 69(2004)011901)

T-indep. term

T2-term Non Fermi-liquid effect

Magnetic susceptibility at T>0

Paramag.

FM

Magnetic phase diagram of QCD

Curie (critical) temperature should be order of several tens of MeV.

Non-Fermi-liquid effect

VI. Summary and concluding remarks

・ We have considered magnetic susceptibility of QCDwithin Fermi-liquid theoryRoles of static and dynamic screening are figured out:

Static

Dynamic

24 ln gg

TT ln2 Novel non-Fermi liquid effect!

・ Non-perturbative effects, Large Nc, Nf etc. other than OGE

・ How to distinguish various ideas about the origin

Thermal evolution as well as magnetic evolution

No decay of B, suggested by statistical analysis of X-ray pulsars

Novel mechanism of emissivity or superconductivity

Spin wave or magnons （ T.T., arxiv:07113349 ）