Ferromagnetic properties of quark matter
description
Transcript of Ferromagnetic properties of quark matter
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 )