Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field
In dense neutron matter under the presence of a strong magnetic field, considered in the model with the Skyrme effective interaction, there are possible two types of spin ordered states. In one of them the majority of neutron spins are aligned opposite to magnetic field (thermodynamically preferable...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1110662017-01-09T03:02:22Z Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field Isayev, A.A. Yang, J. Ядерная физика и элементарные частицы In dense neutron matter under the presence of a strong magnetic field, considered in the model with the Skyrme effective interaction, there are possible two types of spin ordered states. In one of them the majority of neutron spins are aligned opposite to magnetic field (thermodynamically preferable state), and in other one the majority of spins are aligned along the field (metastable state). The equation of state, incompressibility modulus and velocity of sound are determined in each case with the aim to find the peculiarities allowing to distinguish between two spin ordered phases. У густій нейтронній матерії за наявністю сильного магнітного поля в моделі з ефективною взаємодією Скірма можливі два типи спінововпорядкованих станів. В одному з них більшість спінів орієнтовані протилежно магнітному полю (термодинамічно кращий стан), в іншому - вздовж поля. У кожному випадку визначені: рівняння стану, модуль нестисловості та швидкість звуку з метою знайти особливості, що дозволяють відрізнити ці спінововпорядковані стани. В плотной нейтронной материи при наличии сильного магнитного поля в модели с эффективным взаимодействием Скирма возможны два типа спиновоупорядоченных состояний. В одном из них большинство нейтронных спинов ориентировано противоположно магнитному полю (термодинамически предпочтительное состояние), в другом - по полю (метастабильное состояние). В каждом случае определены уравнение состояния, модуль несжимаемости и скорость звука с целью найти особенности, позволяющие отличить эти спиновоупорядоченные фазы. 2011 Article Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field / A.A. Isayev, J. Yang // Вопросы атомной науки и техники. — 2011. — № 3. — С. 3-9 — Бібліогр.: 40 назв. — англ. 1562-6016 PACS: 21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe http://dspace.nbuv.gov.ua/handle/123456789/111066 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы |
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Ядерная физика и элементарные частицы Ядерная физика и элементарные частицы Isayev, A.A. Yang, J. Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field Вопросы атомной науки и техники |
| description |
In dense neutron matter under the presence of a strong magnetic field, considered in the model with the Skyrme effective interaction, there are possible two types of spin ordered states. In one of them the majority of neutron spins are aligned opposite to magnetic field (thermodynamically preferable state), and in other one the majority of spins are aligned along the field (metastable state). The equation of state, incompressibility modulus and velocity of sound are determined in each case with the aim to find the peculiarities allowing to distinguish between two spin ordered phases. |
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Article |
| author |
Isayev, A.A. Yang, J. |
| author_facet |
Isayev, A.A. Yang, J. |
| author_sort |
Isayev, A.A. |
| title |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| title_short |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| title_full |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| title_fullStr |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| title_full_unstemmed |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| title_sort |
spin ordered phase transitions in neutron matter under the presence of a strong magnetic field |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2011 |
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Ядерная физика и элементарные частицы |
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http://dspace.nbuv.gov.ua/handle/123456789/111066 |
| citation_txt |
Spin ordered phase transitions in neutron matter under the presence of a strong magnetic field / A.A. Isayev, J. Yang // Вопросы атомной науки и техники. — 2011. — № 3. — С. 3-9 — Бібліогр.: 40 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT isayevaa spinorderedphasetransitionsinneutronmatterunderthepresenceofastrongmagneticfield AT yangj spinorderedphasetransitionsinneutronmatterunderthepresenceofastrongmagneticfield |
| first_indexed |
2025-07-08T01:34:17Z |
| last_indexed |
2025-07-08T01:34:17Z |
| _version_ |
1837040620789563392 |
| fulltext |
NUCLEAR PHYSICS AND ELEMENTARY PARTICLES
SPIN ORDERED PHASE TRANSITIONS IN NEUTRON
MATTER UNDER THE PRESENCE OF A STRONG
MAGNETIC FIELD
A.A. Isayev 1∗, J. Yang 2†
1National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2Department of Physics and the Institute for the Early Universe,
Ewha Womans University, Seoul 120-750, Korea
(Received February 25, 2011)
In dense neutron matter under the presence of a strong magnetic field, considered in the model with the Skyrme
effective interaction, there are possible two types of spin ordered states. In one of them the majority of neutron spins
are aligned opposite to magnetic field (thermodynamically preferable state), and in other one the majority of spins
are aligned along the field (metastable state). The equation of state, incompressibility modulus and velocity of sound
are determined in each case with the aim to find the peculiarities allowing to distinguish between two spin ordered
phases.
PACS: 21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe
1. INTRODUCTION
Magnetars are strongly magnetized neutron stars [1]
with emissions powered by the dissipation of mag-
netic energy. Magnetars are thought to give the ori-
gin to the extremely powerful short-duration γ-ray
bursts [2, 3]. The magnetic field strength at the sur-
face of a magnetar is about of 1014-1015 G [4, 5].
Such huge magnetic fields can be inferred from ob-
servations of magnetar periods and spin-down rates,
or from hydrogen spectral lines. In the interior of
a magnetar the magnetic field strength may be even
larger, reaching values of about 1018 G [6, 7]. Un-
der such circumstances, the issue of interest is the
behavior of neutron star matter in a strong magnetic
field [6, 7, 8, 9].
In the recent study [9], neutron star matter was
approximated by pure neutron matter in a model
with the effective nuclear forces. It was shown that
the behavior of spin polarization of neutron matter
in the high density region in a strong magnetic field
crucially depends on whether neutron matter devel-
ops a spontaneous spin polarization (in the absence of
a magnetic field) at several times nuclear matter sat-
uration density, or the appearance of a spontaneous
polarization is not allowed at the relevant densities
(or delayed to much higher densities). The first case
is usual for the Skyrme forces [10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21], while the second one is charac-
teristic for the realistic nucleon-nucleon (NN) inter-
action [22, 23, 24, 25, 26, 27, 28]. In the former case,
a ferromagnetic transition to a totally spin polarized
state occurs while in the latter case a ferromagnetic
transition is excluded at all relevant densities and the
spin polarization remains quite low even in the high
density region. If a spontaneous ferromagnetic transi-
tion is allowed, it was shown in the subsequent model
consideration with the Skyrme effective forces [29]
that the self-consistent equations for the spin polar-
ization parameter at nonzero magnetic field have not
only solutions corresponding to negative spin polar-
ization (with the majority of neutron spins oriented
opposite to the direction of the magnetic field) but,
because of the strong spin-dependent medium corre-
lations in the high-density region, also the solutions
with positive spin polarization. In the last case, the
formation of a metastable state with the majority
of neutron spins oriented along the magnetic field is
possible in the high-density interior of a neutron star.
In the present study, we provide the zero-
temperature calculations of the equation of state
(EoS), incompressibility modulus and sound velocity
for neutron matter in a strong magnetic field with
the aim to find the peculiarities allowing to distin-
guish between two possible spin ordered states - the
stable one with negative spin polarization and the
metastable one with positive spin polarization. It
will be shown that in the thermodynamically stable
state the incompressibility modulus and the speed
of sound are characterized by the appearance of the
well-defined maximum just around the density at
which the ferromagnetic (FM) phase transition sets
in. Contrarily to that, such features are missing in
the metastable state. Besides, all calculated quanti-
∗E-mail address: isayev@kipt.kharkov.ua
†E-mail address: jyang@ewha.ac.kr
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2011, N3.
Series: Nuclear Physics Investigations (55), p.3-9.
3
ties behave differently under changing magnetic field
in stable and metastable states.
At this point, it is worthy to note that we consider
thermodynamic properties of spin polarized states in
neutron matter in a strong magnetic field up to the
high density region relevant for astrophysics. Nev-
ertheless, we take into account the nucleon degrees
of freedom only, although other degrees of freedom,
such as pions, hyperons, kaons, or quarks could be
important at such high densities.
2. BASIC EQUATIONS
Here we only outline the basic equations necessary
for further calculations, and a more detailed descrip-
tion of a Fermi-liquid approach to neutron matter
in a strong magnetic field can be found in our ear-
lier work [29]. The normal (nonsuperfluid) states of
neutron matter are described by the normal distrib-
ution function of neutrons fκ1κ2 = Tr %a+
κ2
aκ1 , where
κ ≡ (p, σ), p is momentum, σ is the projection of
spin on the third axis, and % is the density matrix of
the system [18, 19, 21]. Further it will be assumed
that the third axis is directed along the external mag-
netic field H. The self-consistent matrix equation for
determining the distribution function f follows from
the minimum condition of the thermodynamic poten-
tial [30] and is
f = {exp(Y0ε + Y4) + 1}−1 ≡ {exp(Y0ξ) + 1}−1
. (1)
Here the single particle energy ε and the quantity
Y4 are matrices in the space of κ variables, with
Y4κ1κ2 = Y4δκ1κ2 , Y0 = 1/T , and Y4 = −µ0/T be-
ing the Lagrange multipliers, µ0 being the chemical
potential of neutrons, and T the temperature. Given
the possibility for alignment of neutron spins along
or oppositely to the magnetic field H, the normal
distribution function of neutrons and single particle
energy can be expanded in the Pauli matrices σi in
spin space
f(p) = f0(p)σ0 + f3(p)σ3, (2)
ε(p) = ε0(p)σ0 + ε3(p)σ3.
Using Eqs. (1) and (2), one can express evidently
the distribution functions f0, f3 in terms of the quan-
tities ε:
f0 =
1
2
{n(ω+) + n(ω−)}, (3)
f3 =
1
2
{n(ω+)− n(ω−)}.
Here n(ω) = {exp(Y0ω) + 1}−1 and
ω± = ξ0 ± ξ3, (4)
ξ0 = ε0 − µ0, ξ3 = ε3.
As follows from the structure of the distribution
functions f , the quantities ω± play the role of the
quasiparticle spectrum and correspond to neutrons
with spin up and spin down. The distribution func-
tions f should satisfy the normalization conditions
2
V
∑
p
f0(p) = %, (5)
2
V
∑
p
f3(p) = %↑ − %↓ ≡ ∆%. (6)
Here % = %↑ + %↓ is the total density of neutron mat-
ter, %↑ and %↓ are the neutron number densities with
spin up and spin down, respectively. The quantity
∆% may be regarded as the neutron spin order para-
meter. The spin ordering in neutron matter can also
be characterized by the neutron spin polarization pa-
rameter
Π =
%↑ − %↓
%
≡ ∆%
%
.
The spin order parameter determines the magne-
tization of the system M = µn∆%, µn being the
neutron magnetic moment. The magnetization may
contribute to the internal magnetic field B = H +
4πM . However, we will assume, analogously to
Refs. [7, 9], that the contribution of the magnetiza-
tion to the magnetic field B remains small for all
relevant densities and magnetic field strengths, and,
hence, B ≈ H. This assumption holds true due
to the tiny value of the neutron magnetic moment
µn = −1.9130427(5)µN ≈ −6.031 ·10−18 MeV/G [31]
(µN being the nuclear magneton) and is confirmed
numerically in a subsequent integration of the self-
consistent equations.
In order to get the self–consistent equations for
the components of the single particle energy, one has
to set the energy functional of the system. In view of
the above approximation, it reads [19]
E(f) = E0(f,H) + Eint(f) + Efield, (7)
E0(f, H) = 2
∑
p
ε 0(p)f0(p)− 2µnH
∑
p
f3(p),
Eint(f) =
∑
p
{ε̃0(p)f0(p) + ε̃3(p)f3(p)},
Efield =
H2
8π
V,
where
ε̃0(p) =
1
2V
∑
q
Un
0 (k)f0(q), k =
p− q
2
, (8)
ε̃3(p) =
1
2V
∑
q
Un
1 (k)f3(q). (9)
Here ε 0(p) = p 2
2m0
is the free single particle spectrum,
m0 is the bare mass of a neutron, Un
0 (k), Un
1 (k) are
the normal Fermi liquid (FL) amplitudes, and ε̃0, ε̃3
are the FL corrections to the free single particle spec-
trum. Note that the field contribution Efield, being
the energy of the magnetic field in the absence of
matter, leads only to the constant shift of the total
energy and, by this reason, can be omitted. Using
4
Eq. (7), one can get the self-consistent equations in
the form [19]
ξ0(p) = ε 0(p) + ε̃0(p)− µ0, (10)
ξ3(p) = −µnH + ε̃3(p). (11)
To obtain numerical results, we utilize the effective
Skyrme interaction [32]. Expressions for the normal
FL amplitudes in terms of the Skyrme force parame-
ters were written in Refs. [30, 33]. Thus, using expres-
sions (3) for the distribution functions f , we obtain
the self-consistent equations (10), (11) for the com-
ponents of the single-particle energy ξ0(p) and ξ3(p),
which should be solved jointly with the normaliza-
tion conditions (5), (6). Further we do not take into
account the effective tensor forces, which lead to cou-
pling of the momentum and spin degrees of freedom,
and, correspondingly, to anisotropy in the momen-
tum dependence of FL amplitudes with respect to
the spin quantization axis.
If the self-consistent equations have a few
branches of the solutions, it is necessary to compare
the corresponding energies (at zero temperature) in
order to decide which solution is thermodynamically
preferable. The energy per neutron, E/A, can be di-
rectly calculated from Eq. (7). The equation of state
(EoS) of neutron matter in a strong magnetic field
then can be obtained from the equation
P = %2 ∂
(
e/%
)
∂ %
, (12)
where e = %(mc2 +E/A) is the energy density, which
includes also the rest energy term. The incompress-
ibility modulus, K = 9∂P
∂% , according to Eq. (12),
reads
K = 9%2 ∂ 2
(
E/A
)
∂%2
+ 18
P
%
. (13)
The speed of sound, vs = c
√
∂P
∂e , can be related to
the incompressibility modulus by the equation
vs = c
√
K
9(mc2 + E
A + P
% )
. (14)
3. EOS OF DENSE NEUTRON MATTER
IN A STRONG MAGNETIC FIELD
The self-consistent equations were analyzed at zero
temperature in Ref. [29] for the magnetic field
strengths up to Hmax ∼ 1018 G, allowed by a scalar
virial theorem [34], in the model consideration with
SLy4 and SLy7 Skyrme effective forces [35]. It was
shown that a thermodynamically stable branch of
solutions for the spin polarization parameter as a
function of density corresponds to negative spin po-
larization when the majority of neutron spins are
oriented opposite to the direction of the magnetic
field. Besides, beginning from some threshold den-
sity %th ∼ 4%0, being slightly dependent on the mag-
netic field strength, the state with positive spin po-
larization can also be realized as a metastable state
in neutron matter (cf. branches Π1 and Π3 in Fig. 2
of Ref. [29]). This conclusion was based on the com-
parison of the free energies of two states which turn
out to be very close to each other [29, 36]. However,
as it will be shown later, additional constraints, such
as, e.g., stability of the system with respect to the
density fluctuations, will define more accurately the
density range admissible for the state with positive
spin polarization.
In this work, the previous study [29] will be ex-
tended by calculating the EoS of dense neutron mat-
ter in a strong magnetic field for various branches of
solutions of the self-consistent equations. Each pos-
sible state should match the constraint K > 0 for
allowable densities and magnetic field strengths be-
ing the condition of the mechanical stability of the
system. Besides, in the high-density region, the ve-
locity of sound should not exceed the speed of light in
the vacuum, vs < c. Note that further the contribu-
tion of the magnetic field pressure to the total pres-
sure will be omitted because in the magnetic fields
up to 1018 G the magnetic field pressure is still small
compared to the matter pressure in the high-density
region of interest.
First, we present the results of determining the
zero-temperature EoS of neutron matter in a strong
magnetic field at the density region where both stable
and metastable spin ordered states can be realized.
Because the results of calculations with SLy4 and
SLy7 Skyrme forces are very close, here we present
the obtained dependences only for the SLy7 Skyrme
interaction. Fig. 1 shows the pressure of neutron
matter as a function of density for two branches of
spin polarization, stable Π1 and metastable Π3, cor-
responding to negative and positive polarizations, re-
spectively (the branch Π2 with positive spin polar-
ization considered in Ref. [29] has the considerably
larger energy per neutron as compared to the previ-
ous ones).
Fig.1. Pressure vs. density for the branches Π1
(stable) and Π3 (metastable) of spin polarization in
neutron matter with the Skyrme SLy7 interaction at
H = 1018 G
For the branch Π1, the pressure is the increasing
function of the density for all relevant densities,
and, hence, the incompressibility coefficient is always
5
positive. However, for the branch Π3, beginning from
the threshold density %th ≈ 3.92%0 up to the den-
sity %c ≈ 4.85%0 (at H = 1018 G), the pressure de-
creases with the density. Hence, in this density range
the incompressibility coefficient is negative and the
metastable state characterized by the branch Π3 of
positive spin polarization cannot appear at these den-
sities. However, beyond the critical density %c, the
metastable state with positive spin polarization is al-
lowed by the criterion K > 0. Note that the EoS for
the metastable state of neutron matter in a strong
magnetic field is stiffer than that for the thermody-
namically equilibrium state. Fig. 2 shows the pres-
sure of neutron matter as a function of the magnetic
field strength for the branches Π1 and Π3 of spin po-
larization at % = 5%0. It is seen that the dependence
of the EoS for stable and metastable branches is dif-
ferent: for the branch Π1 the EoS becomes softer with
the magnetic field while for the branch Π3 stiffer.
Fig.2. Pressure vs. magnetic field strength for the
branches Π1 and Π3 of spin polarization in neutron
matter with the Skyrme SLy7 interaction at % = 5%0
These calculations show that the impact of the mag-
netic field on the EoS remains small up to the field
strengths of about 1017 G.
Fig.3. Same as in Fig. 1, but for the incompressibil-
ity modulus of neutron matter in a strong magnetic
field
Fig. 3 shows the zero-temperature incompressibility
modulus of neutron matter in a strong magnetic field
as a function of density for the branches Π1 and Π3 of
spin polarization. For the branch Π3, the incompress-
ibility modulus monotonously increases with the den-
sity and changes sign from negative to positive at the
critical density %c, marking the stability range with
respect to density fluctuations at densities beyond
%c. As a consequence, if the metastable state with
positive spin polarization can be realized in the high-
density region of neutron matter in a strong magnetic
field, under decreasing density (going from the inte-
rior to the outer regions of a magnetar) it changes at
the critical density %c to a thermodynamically stable
state with negative spin polarization.
For the branch Π1, the behavior of incompressibil-
ity modulus is nonmonotone. The most important
peculiarity is that just around the density (%tr ≈
3.16%0 at H = 1018 G) at which the magnitude of
the spin polarization parameter for the branch Π1
begins rapidly to increase (cf. Fig. 2 of Ref. [29]),
the increasing behavior of the incompressibility mod-
ulus with the density changes on the decreasing one.
Because the density %tr can be regarded as the den-
sity at which a ferromagnetic state sets in, this qual-
itative feature in the behavior of the incompressibil-
ity modulus can be used as the characteristic of the
density-driven FM phase transition in neutron matter
possessing equilibrium spin polarization. The incom-
pressibility modulus decreases till the density about
4%0 at which the spin polarization parameter is well
developed and gets about two third of its strength.
Then there is the plateau in the density dependence of
incompressibility modulus till the density about 5%0
beyond which the incompressibility modulus begins
gradually to increase. Note that the noticeable de-
crease of the incompressibility modulus around the
density of the transition to the ferromagnetic state
was mentioned also in Ref. [37], although the total
incompressibility modulus was not explicitly shown
there, but only that for the spin-up and spin-down
neutron components in the state with equilibrium
spin polarization. Besides, in Refs. [9, 37], there
were no any calculations related to the metastable
branch of spin polarization because this branch itself
was missed in these studies.
Fig.4. Incompressibility modulus vs. magnetic
field strength for the branches Π1 and Π3 of spin
polarization in neutron matter with the Skyrme
SLy7 interaction at % = 5%0
Fig. 4 shows the incompressibility modulus of neutron
matter as a function of the magnetic field strength
6
for the branches Π1 and Π3 of spin polarization at
the density % = 5%0. For the branch Π1, the incom-
pressibility modulus increases with the magnetic field
strength while for the branch Π3 it decreases. It fol-
lows from these calculations that the impact of the
magnetic field on the incompressibility modulus re-
mains mild up to the field strengths of about 1017 G.
Fig. 5 shows the sound velocity in neutron matter
under the presence of a strong magnetic field as a
function of density for the branches Π1 and Π3 of
spin polarization. For the branch Π3, Eq. (14) auto-
matically guarantees the fulfillment of the condition
K > 0 (for all relevant densities E/A > 0).
Fig.5. Same as in Fig. 1, but for the sound velocity
in neutron matter under the presence of a strong
magnetic field
It is seen that for both branches at the relevant den-
sities the superluminous regime doesn’t occur. While
for the branch Π3 the sound velocity monotonously
increases with the density, for the branch Π1 it has
non-monotone behavior. In fact, near the transition
density %tr the sound velocity in a thermodynami-
cally stable state has a clear peak structure consider-
ably decreasing at the densities where the ferromag-
netic phase sets in. This feature, together with the
presence of the maximum in the density dependence
of the incompressibility modulus, can be used for the
identification of the density-driven FM phase tran-
sition in neutron matter possessing equilibrium spin
polarization. On the other hand, the incompressibil-
ity modulus and the speed of sound monotonously
increase with density in the metastable state with
positive spin polarization, and hence, these features
can be used for distinguishing between the thermo-
dynamically stable (negative spin polarization) and
metastable (positive spin polarization) states in neu-
tron matter under the presence of a strong magnetic
field. Fig. 6 shows the sound velocity in neutron mat-
ter as a function of the magnetic field strength for the
branches Π1 and Π3 of spin polarization at the den-
sity % = 5%0. For the branch Π1, the sound velocity
increases with the magnetic field strength while for
the branch Π3 it decreases. These trends are quite
similar to those in the behavior of the incompress-
ibility modulus K(H) for the branches Π1 and Π3.
Fig.6. Same as in Fig. 4 but for the sound velocity
in neutron matter under the presence of a strong
magnetic field
4. CONCLUSIONS
In dense neutron matter under the presence of a
strong magnetic field, considered in the model with
the Skyrme effective interaction, there are possible
two types of spin ordered states: the one with the
majority of neutron spins aligned opposite to mag-
netic field (thermodynamically preferable state), and
the other one with the majority of spins aligned along
the field (metastable state). The equation of state,
incompressibility modulus and velocity of sound have
been determined in each case for SLy7 Skyrme force
with the aim to find the peculiarities allowing to dis-
tinguish between two spin ordered phases.
For the stable state with the branch Π1 of neg-
ative spin polarization, the EoS is softer than that
for metastable state with the branch Π3 of positive
spin polarization. The condition of the positiveness
of the incompressibility modulus, K > 0, is satisfied
for all relevant densities and magnetic field strengths
for the stable branch Π1. However, for the branch Π3,
although formally the solutions of the self-consistent
equations exist at densities larger than some thresh-
old one, %th, the condition K > 0 is satisfied only
at the densities larger than the critical one, %c (e.g.,
for H = 1018 G, %th ≈ 3.92%0 and %c ≈ 4.85%0). As
a consequence, if the metastable state with positive
spin polarization can be realized in the high-density
region of neutron matter in a strong magnetic field,
under decreasing density (going from the interior to
the outer regions of a magnetar) it changes at the crit-
ical density %c to a thermodynamically stable state
with negative spin polarization.
For the thermodynamically stable branch Π1, the
incompressibility modulus and the speed of sound are
characterized by the appearance of the well-defined
maximum just around the density at which the ferro-
magnetic phase sets in. The last qualitative features
can be used for the identification of the density-driven
FM phase transition in neutron matter, possessing
equilibrium spin polarization, under the presence of
a strong magnetic field. Contrarily to the previous
case, for the branch Π3 of positive spin polariza-
tion, the incompressibility modulus and the speed of
sound monotonously increase with density that can
7
be used to distinguish between two different spin or-
dered phases.
The dependence of all calculated quantities on the
magnetic field strength H turns out to be different for
two spin ordered phases. For the thermodynamically
stable branch Π1, the incompressibility modulus and
sound velocity increase with H while the pressure
decreases. The exactly opposite tendency has been
found for the branch Π3 of positive spin polarization
that also allows one to differentiate between spin po-
larized states with opposite polarizations in neutron
matter under the presence of a strong magnetic field.
As yet one problem for consideration, it would
be interesting to study the role of finite temperature
effects on the EoS, incompressibility modulus and
speed of sound in dense neutron matter in a strong
magnetic field. As has been shown already, these ef-
fects can lead to a number of nontrivial features such
as, e.g., unusual behavior of the entropy in various
spin ordered systems [36, 38, 39, 40].
J.Yang was supported by grant 2010-0011378
from Basic Science Research Program through NRF
of Korea funded by MEST and by grant R32-10130
from WCU project of MEST and NRF.
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ФАЗОВЫЕ ПЕРЕХОДЫ С УПОРЯДОЧЕНИЕМ СПИНОВ В НЕЙТРОННОЙ
МАТЕРИИ ПРИ НАЛИЧИИ СИЛЬНОГО МАГНИТНОГО ПОЛЯ
А.А. Исаев, Дж. Янг
В плотной нейтронной материи при наличии сильного магнитного поля в модели с эффективным
взаимодействием Скирма возможны два типа спиновоупорядоченных состояний. В одном из них боль-
шинство нейтронных спинов ориентировано противоположно магнитному полю (термодинамически
предпочтительное состояние), в другом – по полю (метастабильное состояние). В каждом случае опре-
делены: уравнение состояния, модуль несжимаемости и скорость звука с целью найти особенности,
позволяющие отличить эти спиновоупорядоченные фазы.
ФАЗОВI ПЕРЕХОДИ З УПОРЯДКУВАННЯМ СПIНIВ У НЕЙТРОННIЙ МАТЕРIЇ
ЗА НАЯВНIСТЮ СИЛЬНОГО МАГНIТНОГО ПОЛЯ
О.О. Iсаєв, Дж. Янг
У густiй нейтроннiй матерiї за наявнiстю сильного магнiтного поля в моделi з ефективною взаємодiєю
Скiрма можливi два типи спiнововпорядкованих станiв. В одному з них бiльшiсть спiнiв орiєнтованi
протилежно магнiтному полю (термодинамiчно кращий стан), в iншому – вздовж поля. У кожному ви-
падку визначенi: рiвняння стану, модуль нестисловостi та швидкiсть звуку з метою знайти особливостi,
що дозволяють вiдрiзнити цi спiнововпорядкованi стани.
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