Physics Letters B 751 (2015) 123–126
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Spontaneous magnetization of quark matter in the inhomogeneous
chiral phase
R. Yoshiike
∗
, K. Nishiyama, T. Tatsumi
Department of physics, Kyoto University, Kyoto 606-8502, Japan
a r t i c l e i n f o a b s t r a c t
Article history:
Received
10 July 2015
Received
in revised form 8 October 2015
Accepted
12 October 2015
Available
online 20 October 2015
Editor:
W. Haxton
Keywords:
Quark
matter
Inhomogeneous
chiral phase
Spontaneous
magnetization
Magnetar
Considering the density wave of scalar and pseudoscalar condensates, we study the response of quark
matter to a weak external magnetic field. In an external magnetic field, the energy spectrum of the
lowest Landau level becomes asymmetric about zero, which is closely related to chiral anomaly, and
gives rise to the spontaneous magnetization. This mechanism may be one of candidates for the origin of
the strong magnetic field in pulsars and/or magnetars.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
Recently, the existence of the inhomogeneous chiral phase in
the QCD phase diagram has been discussed by the analysis of the
effective models such as Nambu–Jona–Lasinio (NJL) model [1–3] or
the Schwinger–Dyson approach [4]. In this phase, the quark con-
densates
spatially modulate and it is very similar to the FFLO state
in superconductor [5,6] or spin/charge density wave [7,8]. Here, we
consider “dual chiral density wave (DCDW)” [1] among many kinds
of form of the condensates: the quark condensates then take the
form,
(r) ≡
¯
ψψ+i
¯
ψiγ
5
τ
3
ψ=e
iqz
, (1)
within the two-flavor QCD. This configuration is also obtained by
embedding one of the Hartree–Fock solutions in the NJL
2
model,
so-called chiral spiral [9,10]. Since the DCDW phase has been ex-
pected
to appear in the moderate density region [1], it may be
plausible that this phase is realized in neutron stars.
The
effect of the magnetic field has been first discussed by
Frolov et al. for the DCDW phase [11]. They have found that the
spatial direction of the wavevector q is favored to be parallel to the
magnetic field, and the domain of the DCDW phase is much ex-
tended
in the QCD phase diagram. In Ref. [12] these features arise
from
some topological effect through spectral asymmetry of the
quark energy; quarks exhibit an interesting feature in the presence
*
Corresponding author.
E-mail
address: yoshiike@ruby.scphys.kyoto-u.ac.jp (R. Yoshiike).
of the magnetic field and the energy spectrum becomes asymmet-
ric
about zero. There also appear new terms in the generalized
Ginzburg–Landau expansion due to spectral asymmetry, which sig-
nals
the novel Lifshitz point in the QCD phase diagram. Thus,
they emphasized the peculiar role of the phase degree of freedom
of (r).
Here
we further inquire this issue. We study magnetic prop-
erties
of the DCDW phase to reveal another aspect, spontaneous
magnetization in the DCDW phase, which suggests a microscopic
origin of the strong magnetic field in compact stars.
The
origin of the strong magnetic field in compact stars has
been one of the long-standing problems. In particular, magnetars
have the huge magnetic field ∼ 10
15
Gon the surface [13,14]. As a
candidate of the origin, amplification of the magnetic field by the
dynamo mechanism, magnetorotational instability or the hypothe-
sis
of the fossil magnetic field has been proposed so far from the
macroscopic point of view. Although numerical simulations have
been actively performed, no definite conclusions have been ob-
tained.
From the microscopic point of view, it has been proposed
that the spontaneous magnetization emerges by spin alignment
of quarks on the analogy of the electron gas [15]. However, this
phase should be developed in the low density region. As another
mechanism, it has been proposed that axial anomaly acting on the
parallel layer of the pion domain wall produces magnetization in
nuclear matter [16,17].
We
use the two-flavor NJL model in the mean field approxima-
tion.
It is sufficient to consider the each flavor case because La-
http://dx.doi.org/10.1016/j.physletb.2015.10.028
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.