Topological aspects of the Inhomogeneous chiral phases and implication on compact stars

T. Tatsumi (Kyoto U.)

contents:

I IntroductionII Inhomogeneous chiral phase in the magnetic fieldIII Spectral asymmetry in the inhomogeneous chiral phaseIV Novel tricritical (Lifshitz) point in the presence of the magnetic fieldV Phenomenological implicationsVI Summary and concluding remarks

In collaboration with R. Yoshiike, K. Nishiyama, S. Karasawa and T. Muto

1

2

2nn

n

1710 G

G. Endroedi al., arXiv:1311.0648.A.Bazabov et al., PRD 80(2009) 014504.

I. Introduction

0

qq

qq

QCD phase diagram(For a review , K. Fukushima and T. Hatsuda,Rept. Prog. Phys. 74, 014001 (2011))

Neutron Stars

Heavy-ion collision

12 1510 G

1710 G

QCD vacuum in the magnetic field

2

Generalized chiral order-parameter:

There appear inhomogeneous chiral phases in the vicinity of the chiral transition :

Critical point (CCP) as Lifshitz point (LP)

Dual chiral density wave (DCDW):

Real kink crystal (RKC):

(T. T. and E. Nakano, hep-ph/0408294. E. Nakano and T. T., PRD 71 (2005) 114006.)

(D.Nickel, PRL 103(2009) 072301; PRD 80(2009) 074025.)

(B. Ruester)

Chiral transition

CCP

Compact stars

FM?

QCD phase diagram

DCDW

H

5 3 ( ) exp( ( ))M i q iqq qi r r

Two typical forms of the condensateshave been studied within NJL model:

Chiral phase transition

For a recent review, M. Buballa and S. Carignano, arXiv:1406.1367

order parameter: qq

3

T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006.

2 ,M G

qq

=- D

D =

cBm

DCDW(dual chiral density wave)

qq5qi q τ

MF

0M

Usual chiral transition 　（ T=0)

DCD

W

z

5 3iy g t y

yy

Instability of the Fermi surface due to the ph excitations (Nesting effect)It provides another path of the restoration of chiral symmetry

SSB

4

DCDW

5 3

cos( )

sin( )

qq qz

qi q qz

ref. T.T. and E. Nakano, hep-ph/0408294 PRD71(2005)114006.

5MeVcm

Chiral limit

Tricritical point=Lifshitz point

Critical end point(CEP)

S. Karasawa and T.T., arXiv:1307.6448.

m

Results within 2 flavor NJL model

SSB

restored

DCDW

5

Inhomogeneous (spatially dependent) order parameterin the vicinity of the phase transition

FFLO state in superconductorSpin density wave (SDW) from ferromagnetic phase

( ) ( 0) r( ) ( 0) M M r

Chiral symmetry (SSB) ˆ ˆ ( ) ( 0) qOq qOq r

General solutions of the one dimensional order G.Basar and G.V.Dunne, PRL 100(2008)2004004; PRD 78(2008) 065022.

Twisted kink crystal condensate

5 3qi q

z

qq

Common feature in condensed matter physics ?!

6

II Inhomogeneous chiral phase in the magnetic field

Phase transition・ Enhancement of SSB or Magnetic catalysis

S. P. Klevansky and R.H. Lemmer, PRD 39 (1989) 3478.H. Suganuma and T.T., Ann.Phys. 208 (1991) 371.

V.P. Gusynin, V.A. Milansky, I.A. Shovkovy, NPB 462 (1996) 249

q q e e

H

Cooper pair

E

B

S.P. Klevansky , RMP 64(1992) 649.

7

(0,0, )HHDirac operator in the magnetic field:

(0, ,0)HxADirac Hamiltonian with the inhomogeneous condensate:

* †5 3 5 31 1( ) ( ) ,

2 2D DH M z M z H i e

α P P A

Using the Landau gauge,

( ) 2 exp( )M z G iqz （ DCDW ）

DCDW

restored

m

(eH

)1/

2

T=0

1610 G

nucl(3 5) H=0I.E. Frolov et al, PRD 82, 076002 (2010)

Hybrid condensate

with mod

2

u

2( ) sn ; exp( )

1 1

lus 0< 1.

m mzM z iqz

RKC

Competition between DCDW and RKCNishiyama’s talk in Session 6

8

Energy spectrum :

Spectral asymmetry

2 2

0,

22 2

, ,

(

( ) / 2 2 , 1, 2,3,...

) / 2, (Lowest Landau level(LLL))n

nE p m p q eHn n

E p m p q

[I.E. Frolov et al., Phys. Rev. D82 (2010) 076002]

Cf: ( )2 22 1 , 0,1, 2,..., pE m eH n s p ne e= + + + + =

22 2 2

, , / 2p zE p m p q Spectrum is symmetric about 0

Landau levels:

DCDW

ex) In the case of DCDW ( ) 2 exp( )M z G iqz

III Spectral asymmetry in the inhomogeneous chiral phase

q/2 mm

2( / 2) 2m q eH 0

2( / 2) 2m q eH

LLL1

1

9

It is known that spectral asymmetry induces the anomalous baryon number.

†1ˆ ( ), ( ) .2

Baryon number operator is defined by

DN d x x x Atiyah-Patodi-Singer h invariant (Atiyah et al, Proc.Cambridge Phils. Soc. 77(1975) 42;78(1975) 405;79 (1976) 71):

0

0

0

1 1

lim s i gn ) (

H

s

s

E E

E E

(Goldstone & Jaffe, PRL 51(1983) 1518.)e.g. Chiral bag MIT bag

Skyrmion

Baryon number in the thermodynamic limit:, with fixed / .N V N V

( ) ( )

( ) ( )1s i gn ,

2 1 1H E E

E EN E

e e

Topological one

(A.J. Niemi and G.W. Semenoff, Phys. Reports 135 (1986) 99. 　 )

which must be consistent with the thermodynamic relation,/ .N

(“Baryon number of the vacuum”)

10

Using the Mellin transform

1

0

2

1

2 1

0

1 32 1 F 1 , , ;

( ) 2

1exp ,

( )

( ) sign( )exp( ) 2

2 2 2 2 2 2

2

s

s s

sH

sq V s s s s qm

s m

E d Es

V dps d E E

s

eH

eH

Gauss hypergeometric func

removable singularity at

tion

0s

0lim ( ) .

2H Hs

eH qs V

h is independent of m.

Direct evaluation of h H for DCDW (LLL only)

q/2

E+=q/2+m

E-=q/2-m

( )E p

０

m

Thus topological baryon-number density can be written as anomalyB ,

2 24for / 2 , / 2

H

eH

Lq

eH q

m q m

LLL

( / 2 )q m

11

Important implications of the spectral asymmetry

・ It is closely related to chiral anomaly・ It induces the anomalous quark number, and broadens the inhomogeneous phase Nishiyama’s talk in Session 6・ It gives a new tricritical (Lifshitz) point on the QCD phase diagram・ It provides us with a possibility of the spontaneous magnetization Yoshiike’s talk

Anomaly in quark matterIt well-known that (chiral) anomaly plays an important role in hadron mass spectrum, hadron dynamics such as

0 2 ,

or hadron models such as skyrmion.However its role in quark matter or many-body dynamics has not been well-understood yet.

D.T. Son and A.R. Zhitnitsky, PRD 70, 074018 (2004).D.T. Son and M.A. Stephanov, PRD 77, 014021 (2008)

12

For the DCDW-type configuration ( )( ) ( ) i zM z m z e

2

0

0

2 2

The invariant can be easily evaluated:

,

Using the derivative expansion,

2( ) cos t

we find

1lim ( ) '( )

4

r

.

Accordingly we ha e

2

c

v

Da

s D L

DH H

s

LLH

o

L

m

L L

n

Hs s d d x

ed x

d

zH

s x

z

z

H Gauging WZW action .f.

Spontaneous Magnetization?

2'

4z

eM

(D.T. Son and M.A. Stephanov, PRD 77(2008) 014021.)

Yoshiike’s talk

LLLS

5 ( ) / 2 γ r

Chiral anomaly and DCDW(T.T.,K. Nishiyama and S. Karasawa, EPJ Web of Cof. 71, 00131 (2014); arXiv:1405.2155)

13

Generalized Ginzburg-Landau (gGL ) theory:

IV Novel tricritical point in the presence of the magnetic field

The coefficients are evaluated in terms of the Green function in the presence of the magnetic field [Schwinger, (1951)],

2

2 20 0

2 2 2 0 0 3 3 1 20 30

3

2

( , )exp ( 1)

|

tan( )( ) exp ( ) 1 tan 1 tan

,| 2

n n

n

A

eHsS k ds is k k k k e

D eH k

Hs eHse

ie k k e

s

H H n

H

k

k

k γ

(A. Chodos et al (1990))

(sum over the Landau levels)

14

If spectrum is symmetric, the “odd” terms vanish:0, for i=odd.i

Then the Lifshitz point is given by the conditions:

2 4 0, which meets the tricritical point of the chiral transition in the chiral limit [Nickel, (2009)].

The “odd” terms survive in the presence of the magnetic field, due to the spectral asymmetry of the LLL levels.So, we can define the new tricritical point (Lifshitz point) by

2 1nq

2 3 0

Relation between “odd” terms and chiral anomaly?

The odd terms arise as a consequence of the asymmetric appearance of M,M* in the Hamiltonian of the each sector of the Landau levels.

15

q/2

E+=q/2+m

E-=q/2-m

( )E p

０

mNear the Lifshitz point, m,q are small and we can assume without loss of generality.

( ) iqzM z me

,q m

2 22

22

22 2 2

2/ 2

2

/

2

4

4

...8

eHq

eHm q

N V

eH

m

m

q

eH

eH

topological non-topological

2to 3to

Using the thermodynamic relation,

N

we can see the spectral asymmetry of LLL gives “odd” terms; anomaly effect is already included in the “odd” terms.

LLL

16

T

m0

0.1

0.05

0.15

3( )H

3/2

3 3

(

/2

1)

2

3

2

1 1

1( ; , ) 1 2

2 2

0, as Im2 2

0.2

f

cf m

f

f

m

c

e H dkH T N T

i k

e HN i

T

(higher Landau levels don’t contribute to a3 ）

3Thus 0 0

(no divergence)

(tri-gamma function)

LLL contribution to 3

This may be natural to consider that DCDW or pseudoscalar condensate must vanish at m=0 due to parity symmetry, which means the spectral asymmetry also vanishes there.

17

2 includes some divergence, and we regularize it by the proper-time method.

22 4 3, , , , 4 , ,bT H T H T H

Phase diagram in the m-T-H space

Usual chiral transition

Lifshitz point2 ( 0, , ) 0T H

Lifshitz line

which includes contributions from higher Landau levels as well as LLL.

H

1610 G

18

V Phenomenological implications

(i) Spontaneous magnetization

Magnetars

Origin:(i) Fossil field(ii) Dynamo scenario (crust)

(iii)Microscopic origin (core)

Strong magnetic field in compact stars

One of the long standing but very important problems in physicsSince the first discovery of pulsar!

cf T.T., PLB 480, 280 (2000)

Spontaneous magnetization in DCDW? 19

Cas A

(ii) Cooling of hybrid stars

CAS A

3C58

Vela

Suggest nucleon superfluidity?20

†

†1 121 2 1 2 2cos . .

, , | | , , | |

exp( ), 2

W W DCDW W DCVDW

FC

W W W

DCDW i DCDW

W

i i i

u e H d u e H d

h l h

H H U H U

Gh c hU iU h

q r

T. Muto and TT, PTP 80(1988)28.)

The DCDW state can be represented as a chirally rotated state

Rapid cooling mechanism (TT and T. Muto, PRD89, 103005 (2014) .)

“pole” contribution

d

u

een

DCDW ±q

DCDW catalyzed cooling

,which means DCDW modifies the momentum conservation at the weak-interaction vertex.

cf pion cooling (O.Maxwell et al(MBCDM), Ap.J.216(1977)77

Usually neutrino emission through the quark beta decay

is strongly prohibited in neutron or hybrid stars, while it may give a large emission rate.

,e ed u e u e d

21

・ Emissivities are large in both cases, comparable with quark Durca or pion cooling

・ This mechanism works in the limited density region

Cooling of neutron stars

General formula is very complicated,but we can easily estimate the emissivities near the phase boudaries:(i) Near the onset density(ii) Near the termination density

24 26 69

-3 -1(ergs c0 )1 1 m s0DCDW T

22

VI Summary and concluding remarks:・ We have studied dual chiral density wave (DCDW) in the presence of the magnetic field.

・ Spectral asymmetry arises in the LLL and anomalous baryon number is induced.

DCDW phase is greatly extended due to the magnetic field.

・ Generalized Ginzburg-Landau analysis suggests, that there emerges a novel tricritical point 　（ line ） on the H-T plane (m=0).

It should be interesting to explore it in the lattice QCD simulations, free from the sign problem.

m

T

HDCDW

( )cT H

Realistic study is needed to include the current mass or mass dependence.

23

・ Phenomenological implications (i) Spontaneous magnetization is possible? origin of magnetic field in compact stars (ii) Novel cooling mechanism works there and emissivity is almost the same as the pion cooling or quark cooling

・ Stability of the one dimensional order

Landau-Peierls theorem Quasi long range order (QLRO)

Long range orderdue to the spectral asymmetry of LLL

Magnetic field

Change of the dispersion relation of the NG mode2 2 4 2 2 2' ( )z zak bk a k c B k

anisotropic but quadratic in momentum!24