# Hot and highly magnetized neutron star matter properties with Skyrme
interactions

Omar G. Benvenuto1,2111Member of the Carrera del Investigador Científico,
Comisión de Investigaciones Científicas de la Provincia de Buenos Aires, La
Plata, Argentina., Eduardo Bauer1,3 and Isaac Vidaña4 1Facultad de Ciencias
Astronónicas y Geofísicas, Universidad Nacional de la Plata, Paseo del Bosque
S/N, B1900FWA La Plata, Argentina  2Instituto de Astrofísica de la Plata, CCT-
CONICET-UNLP, Argentina  3IFLP, CCT-La Plata CONICET, Argentina  4Istituto
Nazionale di Fisica Nucleare, Sezione di Catania, Dipartimento di Fisica e
Astronomia “Ettore Majorana”, Università di Catania, Via Santa Sofia 64,
I-95123 Catania, Italy

###### Abstract

We study the properties of hot and dense neutron star matter under the
presence of strong magnetic fields using two Skyrme interactions, namely the
LNS and the BSk21 ones. Asking for $\beta$–stability and charge neutrality, we
construct the equation of state of the system and analyze its composition for
a range of densities, temperatures and magnetic field intensities of interest
for the study of supernova and proto-neutron star matter, with a particular
interest on the degree of spin-polarization of the different components. The
results show that system configurations with larger fractions of spin up
protons and spin down neutrons and electrons are energetically favored over
those with larger fractions of spin down protons and spin up neutrons and
electrons. The effective mass of neutrons and protons is found to be in
general larger for the more abundant of their spin projection component,
respectively, spin down neutrons and spin up protons. The effect of the
magnetic field on the Helmhotz total free energy density, pressure and
isothermal compressibility of the system is almost negligible for all the
values of the magnetic field considered.

##  I Introduction

The properties of neutron stars PharosBook  can be drastically modified due to
the presence of strong magnetic fields. Most radio pulsars and accreting
neutron stars in X-ray binaries present surface magnetic fields with
intensities in the range $10^{12}-10^{13}$ G ShapiroTeukolsky . Also recycled
millisecond pulsars and old neutron stars in low-mass X-ray binaries have high
surface fields of about $10^{8}-10^{9}$ G Lewin95 ; Lyne98 . In the surface of
soft-gamma-ray repeaters and slowly spinning anomalous X-ray pulsars, the so-
called “magnetars”, the field can reach values of the order of
$10^{14}-10^{15}$ G Duncan92 ; Paczynski92 ; Thompson95 ; Thompson96 . The
intensity of these fields may grow by several orders of magnitude in the dense
interior of all these compact objects up to an upper limit, $B\leq
10^{18}(M/1.4M_{\odot})(R/10\,\mbox{km})^{-2}$ G, which follows from the
virial theorem of magnetohydrostatic equilibrium ShapiroTeukolsky ;
Chandrasekhar53 . However, the origin of such large intensities remains still
uncertain. These strong fields could be the fossil remnants from those of the
progenitor star Tatsumi06 , or alternatively, they could be generated after
the formation of the neutron star by some kind of dynamo process due to some
long-lived electric currents flowing in the highly conductive neutron star
material Thompson93 . Another possibility that has long been considered by
many authors, with however contradictory results, is that these fields result
from a spontaneous phase transition to a ferromagnetic state at densities
corresponding to theoretically stable neutron stars (see, e.g., Refs.
Brownell69 ; Rice69 ; Clark69 ; Silvertein69 ; Clark69b ; Ostgaard70 ;
Pearson70 ; Pandharipande72 ; Backman73 ; Haensel75 ; Jackson82 ; Vidaurre84 ;
Kutschera89 ; Marcos91 ; Kutschera94 ; Bernardos96 ; Fantoni01 ; Vidana02 ;
Vidana02b ; Rios05 ; Lopez06 ; Bombaci06 ; Sammarruca07 ; Perez09 ; Perez09b ;
Bigdeli10 ; Sammarruca11 ). Whatever is the origin of these fields, however,
it is clear that the study of nuclear matter under the influence of strong
magnetic fields is fundamental for a complete understanding of the magnetic
properties of neutron stars.

Several authors have studied the magnetization of symmetric nuclear matter and
pure neutron matter Khalilov02 ; PerezGarcia08 ; Aguirre11 ; PerezGarcia11 ;
Aguirre13 ; Aguirre14 ; Aguirre15 . The magnetization of $\beta$-stable
neutron star matter, however, has received less attention in the literature.
Blandford and Hernquist in Ref. Blandford82 , for instance, studied
extensively the magnetization of $\beta$-stable matter for a single-component
electron gas and for the crust matter of neutron stars. This study was
generalized years later by Broderick et al. in Ref. Broderick00  by including
also the contribution of neutrons and protons. The effect of the density
dependence of the nuclear symmetry energy on the magnetization of
$\beta$-stable matter was studied few years ago by Dong et al. Dong13 ,
concluding that the magnetic susceptibility of protons, electrons and muons
can be larger than that of neutrons. These authors found also that the
anomalous magnetic moment of protons enhances their magnetic susceptibility to
the point that it can be one of the main contributions and, therefore, should
not be neglected. In Ref. Rabhi15  Rabhi et al. employed a relativistic mean
field (RMF) approach to study the effects of strong magnetic field on the
proton and neutron spin polarization, and the magnetic susceptibility in
asymmetric nuclear matter. The authors of this work showed that magnetic
fields of about $10^{16}-10^{17}$ G have noticeable effects on the range of
densities of interest for the study of neutron star crusts. They found also
that, although the magnetic susceptibility of protons is lager for weaker
fields, the one of neutrons becomes of the same order or even larger at
subsaturation densities for small values of the proton fraction when the
fields are larger than $\sim 10^{16}$ G. In the recent years, the Coimbra
group has published several works devoted to the study the effects of strong
magnetic fields on the crust-core transition and the inner crust of neutron
stars Fang16 ; Fang17 ; Fang17b ; Avancini18 ; Sengo20 ; Pais21 ; Ferreira21 .
In these works the Vlasov equation is used to determine, using a RMF approach,
the dynamical spinodal instability region which gives a good estimation of the
crust-core transition in neutron stars. The results of these works show that
strong magnetic fields of the order of $10^{15}-10^{17}$ G have a large effect
on the spinodal region, defining the crust-core transition as a succession of
stable and unstable regions due to the opening of new Landau levels. The
results of these studies show also that sufficiently strong magnetic fields
can significantly modify the extension of the unstable region and, therefore,
the crust of magnetized neutron stars. The effect of temperature on the crust-
core transition of magnetar was studied by this group in Ref. Fang17b  and
very recently also in Ref. Ferreira21 . In these works, the authors showed
that the effect on the extension of the crust-core transition is washed away
for temperatures above $10^{9}$ K for magnetic field intensities $\sim 5\times
10^{16}$ G but may still persist if a magnetic field as high as $\sim 5\times
10^{17}$ G is considered. They found that for lower temperatures, the effect
of the magnetic field on the crust-cross transition is noticeable and grows as
the temperature decreases.

In this work we study the properties of an electrically neutral system of
neutrons, protons and electrons in equilibrium with respect to the weak
interaction ($\beta$-equilibrium), at finite temperature and in the presence
of strong magnetic fields using Skyrme interactions. We consider a range of
densities, temperatures and magnetic field intensities of interest for the
study of supernova and proto-neutron star matter. We construct the equation of
state (EoS) of the system and analyze its composition, with a particular
interest on the degree of spin-polarization of the different components. The
isothermal compressibility is also calculated and analyzed. The final scope of
this work is to establish a framework for a future study of neutrino
propagation in hot dense neutron star matter under the presence of strong
magnetic fields. Neutrino cross sections and, consequently, the neutrino mean
free path can be substantially affected by the presence of strong magnetic
fields in neutron stars. For instance, the emission of neutrinos becomes
asymmetric. It depends on the direction of the neutrino, as it was recently
shown in Ref. TorresPatino19  where the neutrino mean free path in hot pure
neutron matter under the presence of strong magnetic fields was analyzed by
two of the authors of the present work. The extension of this analysis for the
propagation of neutrinos in (more realistic) $\beta$-stable matter is left for
the near future, and it will be based on the results of the present work.

The paper is organized in the following way. The formalism employed to
determine the properties of hot neutron star matter under the presence of a
external magnetic field is described in Sec. II. Results are presented and
discussed in Sec. III. Finally, a short summary and the main conclusions of
this work are given in Sec. IV.

##  II Formalism

As said in the introduction, in this work we consider an electrically neutral
system of neutrons, protons and electrons in $\beta$-equilibrium, at finite
temperature and in the presence of strong magnetic fields. The physical state
of the system can be obtained by minimizing a function $F$ which is
constructed from the Helmhotz total free energy density ${\cal F}$ of the
system and two constraints that express, respectively, the conditions of
baryon number conservation and electrical charge neutrality

$F={\cal
F}+\alpha\left(\rho-\sum_{\tau=n,p}\sum_{\sigma=\uparrow,\downarrow}\rho_{\tau\sigma}\right)+\beta\sum_{\sigma=\uparrow,\downarrow}\left(\rho_{p\sigma}-\rho_{e\sigma}\right)\
.$ (1)

Here $\rho$ is the total baryon number density, and $\rho_{\tau\sigma}$ and
$\rho_{e\sigma}$ are, respectively, the densities of neutrons ($\tau=n$),
protons ($\tau=p$) and electrons ($e$), with spin up ($\sigma=\,\uparrow$) or
spin down ($\sigma=\,\downarrow$) projections. The quantities $\alpha$ and
$\beta$ are the two Lagrange multipliers associated to each one of the
constraints. The minimization of $F$ requires its partial derivatives with
respect to the particle densities and the two multipliers to be zero, i.e.,

$\displaystyle\frac{\partial F}{\partial\rho_{\tau\sigma}}$ $\displaystyle=$
$\displaystyle 0\ ,\,\,\frac{\partial F}{\partial\rho_{e\sigma}}=0\ ,$
$\displaystyle\frac{\partial F}{\partial\alpha}$ $\displaystyle=$
$\displaystyle 0\ ,\,\,\,\,\,\,\frac{\partial F}{\partial\beta}=0\ .$ (2)

Remembering that the chemical potential of a particle $i$ is
$\mu_{i}=\partial{\cal F}/\partial\rho_{i}$, the above conditions yield the
following set of eight equations

$\mu_{i}-b_{i}\alpha+q_{i}\beta=0,\,\,\,i=n_{\uparrow},n_{\downarrow},p_{\uparrow},p_{\downarrow},e_{\uparrow},e_{\downarrow}\
,$ (3)
$\rho=\sum_{\tau=n,p}\sum_{\sigma=\uparrow,\downarrow}\rho_{\tau\sigma}\ ,$
(4)
$\sum_{\sigma=\uparrow,\downarrow}\rho_{p\sigma}=\sum_{\sigma=\uparrow,\downarrow}\rho_{e\sigma}\
,$ (5)

where $b_{i}$ is the baryon number of particle $i$ and $q_{i}$ its electric
charge. Eliminating the Lagrange multipliers $\alpha$ and $\beta$, one can
obtain a set of relations among the chemical potentials of the different
particles. In general there are as many independent chemical potentials as
there are conserved charges, and all the others can be written in terms of
them. In the case of neutron stars there are only two conserved charges
(baryon number and electric charge), and we chose $\mu_{n_{\uparrow}}$ and
$\mu_{e_{\downarrow}}$ as the two independent chemical potentials associated
with them. Applying now Eq. (3) to $n_{\uparrow}$ and $e_{\uparrow}$ one finds

$\alpha=\mu_{n_{\uparrow}}\ ,\,\,\,\beta=\mu_{e_{\uparrow}}\ .$ (6)

Therefore, we can write

$\mu_{i}=b_{i}\mu_{n_{\uparrow}}-q_{i}\mu_{e_{\uparrow}}\
,\,\,\,i=n_{\uparrow},n_{\downarrow},p_{\uparrow},p_{\downarrow},e_{\uparrow},e_{\downarrow}\
,$ (7)

which together with Eqs. (4) and (5) allow to determine the composition of the
system. Note that

$\mu_{n_{\uparrow}}=\mu_{n_{\downarrow}}\
,\,\,\,\,\mu_{p_{\uparrow}}=\mu_{p_{\downarrow}}\
,\,\,\,\,\mu_{e_{\uparrow}}=\mu_{e_{\downarrow}}\ ,$ (8)

that is, in the physical state the chemical potential of each species is
independent of their spin projection.

We consider the interaction of electrons only with the magnetic field, but not
between electrons with themselves or with protons, whereas to describe the in-
medium interactions among the nucleons we employ Skyrme forces. In particular,
we use the LNS interaction developed by Cao et al., Cao06  and the interaction
BSk21 Goriely10  of the Brussels–Montreal group. We note that the BSk21
interaction contains two new terms, in addition to the usual ones of the
Skyrme force, that are introduced in order to avoid the appearance of
ferromagnetic instability at high densities, a general feature of all the
conventional Skyrme forces developed in the past, as it is the case of the LNS
one.

The total energy density of the system is given by

${\cal E}={\cal E}_{nucl}+{\cal
E}_{elec}+\mu_{N}B\left(2L_{p}+\rho_{p}-\frac{g_{p}}{2}W_{p}-\frac{g_{n}}{2}W_{n}\right)\
,$ (9)

where ${\cal E}_{nucl}$ is the nuclear contribution, obtained in our case
using the Hartree–Fock approximation with the Skyrme interaction, ${\cal
E}_{elec}$ is the electron one, and the last term shows the explicit
dependence of the energy density on the magnetic field. In the next we present
separately these three contributions. We note that throughout all this work we
use natural units in which $\hbar=c=1$.

The nuclear contribution can be written in a compact form as

${\cal
E}_{nucl}=\sum_{\tau=n,p}\sum_{\sigma=\uparrow,\downarrow}\frac{K_{\tau\sigma}}{2m^{*}_{\tau\sigma}}+\frac{1}{16}\left[(a_{0}+a_{2}w^{2})\rho^{2}+a_{1}\left(\sum_{\tau=n,p}W_{\tau}\right)^{2}+a_{3}\left(\sum_{\tau=n,p}I_{\tau}W_{\tau}\right)^{2}\right]\
.$ (10)

Here

$w=\frac{1}{\rho}\left[\sum_{\sigma=\pm 1}\rho_{n\sigma}-\sum_{\sigma=\pm
1}\rho_{p\sigma}\right]$ (11)

is the isospin asymmetry, with

$\rho_{n\sigma}=\frac{1}{(2\pi)^{3}}\int
d^{3}\vec{p}f_{n\sigma}(\varepsilon_{n\sigma}(p),T)$ (12)

and

$\rho_{p\sigma}=\frac{eB}{(2\pi)^{2}}\sum_{N_{p}}\int_{-\infty}^{\infty}dp_{z}f_{p\sigma}(\varepsilon_{p\sigma}(p_{z},N_{p}),T)$
(13)

being the partial densities of neutron and protons with spin projection
$\sigma$. The effective mass of a spin up or down nucleon is given by

$\displaystyle m^{*}_{\tau\sigma}$ $\displaystyle=$
$\displaystyle\Big{[}\frac{1}{m_{\tau}}+\frac{1}{4}(b_{0}+c_{0}-(b_{2}+c_{2})wI_{\tau})\rho$
(14) $\displaystyle+$
$\displaystyle\frac{s_{\sigma}}{4}\sum_{\tau^{\prime}=n,p}(b_{1}+c_{1}+(b_{3}+c_{3})I_{\tau}I_{\tau^{\prime}})W_{\tau^{\prime}}\Big{]}^{-1}\
,$

where $m_{\tau}$ is the bare mass of the nucleon, $I_{\tau}=1(-1)$ for protons
(neutrons), and $s_{\sigma}=1(-1)$ if the spin projection is up (down). The
coefficients $a_{0},\cdot\cdot\cdot,a_{3}$, $b_{0},\cdot\cdot\cdot,b_{3}$ and
$c_{0},\cdot\cdot\cdot,c_{3}$ are given in terms of the parameters
$t_{i=0,\cdot\cdot\cdot,5},x_{i=0,\cdot\cdot\cdot,5}$, $\gamma$ and $\beta$ of
the LNS and BSk21 interactions through the relations

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle
6t_{0}+t_{3}\rho^{\gamma}$ $\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle-2t_{0}(1-2x_{0})-\frac{t_{3}}{3}(1-2x_{3})\rho^{\gamma}$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle-2t_{0}(1+2x_{0})-\frac{t_{3}}{3}(1+2x_{3})\rho^{\gamma}$
$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle-2t_{0}-\frac{t_{3}}{3}\rho^{\gamma}$ $\displaystyle b_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{2}[3t_{1}+t_{2}(5+4x_{2})]$
$\displaystyle b_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[t_{2}(1+2x_{2})-t_{1}(1-2x_{1})]$ $\displaystyle
b_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[t_{2}(1+2x_{2})-t_{1}(1+2x_{1})]$ $\displaystyle
b_{3}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(t_{2}-t_{1})$ $\displaystyle
c_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[3t_{4}\rho^{\beta}+t_{5}\rho^{\gamma}(5+4x_{5})]$
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[t_{5}\rho^{\gamma}(1+2x_{5})-t_{4}\rho^{\beta}(1-2x_{4})]$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[t_{5}\rho^{\gamma}(1+2x_{5})-t_{4}\rho^{\beta}(1+2x_{4})]$
$\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(t_{5}\rho^{\gamma}-t_{4}\rho^{\beta})\ .$ (15)

We note that the terms with the parameters $t_{4},t_{5},x_{4},x_{5}$ and
$\beta$ are absent in the case of the LNS force and, therefore, the
coefficients $c_{0},\cdot\cdot\cdot,c_{3}$ are set equal to zero in this case.
The quantities $K_{\tau\sigma}$ and $W_{\tau}$ appearing in Eqs. (9), (10) and
(14) are related, respectively, with the kinetic energy and the spin asymmetry
density, and are defined as

$K_{n\sigma}=\frac{1}{(2\pi)^{3}}\int
d^{3}\vec{p}p^{2}f_{n\sigma}(\varepsilon_{n\sigma}(p),T)$ (16)
$W_{n}=\rho_{n\uparrow}-\rho_{n\downarrow}$ (17)

in the case of neutrons, and

$K_{p\sigma}=\frac{eB}{(2\pi)^{2}}\sum_{N_{p}}\int_{-\infty}^{\infty}dp_{z}p_{z}^{2}f_{p\sigma}(\varepsilon_{p\sigma}(p_{z},N_{p}),T)$
(18) $W_{p}=\rho_{p\uparrow}-\rho_{p\downarrow}$ (19)

for protons. Note that when the magnetic field is assumed to be along the
direction of the z-axis, due to the Landau quantization, the $x$ and $y$
components of the proton momentum spread over a bound region of area $2\pi eB$
in the $p_{x}-p_{y}$ plane, whereas the $z$ one is not bound and varies
continuously. Therefore, the contribution of the protons to any macroscopic
quantity per unit volume is evaluated by means of the replacement $\int
d^{3}\vec{p}/(2\pi)^{3}\rightarrow eB\int dp_{z}/(2\pi)^{2}\sum_{N_{p}}$. The
sums over the proton Landau levels $N_{p}$, in Eqs. (18) and (19), run from 0
up to a maximum level which is determined numerically, as it is explained with
detail in the appendix B of Ref. Bauer20 .

The functions $f_{n\sigma}(\varepsilon_{n\sigma}(p),T)$ and
$f_{p\sigma}(\varepsilon_{p\sigma}(p_{z},N_{p}),T)$ in Eqs. (12)-(13) and
(16)-(19) are the corresponding neutron and proton Fermi–Dirac momentum
distributions

$f_{\tau\sigma}(\varepsilon_{\tau\sigma},T)=\left[1+\mbox{exp}\left(\frac{\varepsilon_{\tau\sigma}-\mu_{\tau\sigma}}{T}\right)\right]^{-1}\
,$ (20)

where the neutron and proton single-particle energies are, respectively

$\varepsilon_{n\sigma}(p)=m_{n}+\frac{p^{2}}{2m^{*}_{n\sigma}}+\frac{1}{8}v_{n\sigma}-\mu_{N}g_{n}\frac{s_{\sigma}}{2}B\
,$ (21)

and

$\displaystyle\varepsilon_{p\sigma}(p_{z},N_{p})$ $\displaystyle=$
$\displaystyle
m_{p}+\frac{p^{2}_{z}}{2m^{*}_{p\sigma}}+\frac{1}{8}v_{p\sigma}$ (22)
$\displaystyle+$
$\displaystyle\mu_{N}B\left(2N_{p}+1-g_{p}\frac{s_{\sigma}}{2}\right)\ .$

Here $\mu_{N}=3.15245\times 10^{-18}$ MeV/G is the nuclear magneton,
$g_{n}=-3.826$ and $g_{p}=5.586$ are, respectively, the neutron and proton
g-factors which take into account their anomalous magnetic moments, and

$\displaystyle v_{\tau\sigma}$ $\displaystyle=$
$\displaystyle(a_{0}-a_{2}wI_{\tau})\,\rho+s_{\sigma}\sum_{\tau^{\prime}=n,p}(a_{1}+a_{3}I_{\tau}I_{\tau^{\prime}})W_{\tau^{\prime}}$
$\displaystyle+$
$\displaystyle\sum_{\tau^{\prime}=n,p}\sum_{\sigma^{\prime}=\uparrow,\downarrow}(b_{0}+c_{0}+(b_{2}+c_{2})I_{\tau}I_{\tau^{\prime}})K_{\tau^{\prime}\sigma^{\prime}}$
$\displaystyle+$ $\displaystyle
s_{\sigma}\sum_{\tau^{\prime}=n,p}\sum_{\sigma^{\prime}=\uparrow,\downarrow}s_{\sigma^{\prime}}(b_{1}+c_{1}+(b_{3}+c_{3})I_{\tau}I_{\tau^{\prime}})K_{\tau^{\prime}\sigma^{\prime}}$
(23)

is the Skyrme single-particle potential energy.

The electron contribution to the total energy density is

$\displaystyle{\cal E}_{elec}$ $\displaystyle=$
$\displaystyle\frac{eB}{(2\pi)^{2}}\sum_{N_{e}}\sum_{\sigma=\uparrow,\downarrow}$
(24) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dp_{z}\varepsilon_{e\sigma}(p_{z},N_{e})f_{e\sigma}(\varepsilon_{e\sigma}(p_{z},N_{e}),T)\
,$

where, $f_{e\sigma}(\varepsilon_{e\sigma}(p_{z},N_{e}),T)$ is the Fermi–Dirac
distribution of electrons and $\varepsilon_{e\sigma}(p_{z},N_{e})$ their
single-particle energy which reads

$\varepsilon_{e\sigma}(p_{z},N_{e})=\sqrt{m_{e}^{2}+2m_{e}\mu_{B}B(2N_{e}+1-g_{e}\frac{s_{\sigma}}{2})+p_{z}^{2}}\
,$ (25)

with $m_{e},N_{e},\mu_{B}=5.78838\times 10^{-15}$ MeV/G and $g_{e}=-2$ being,
respectively, the mass, the Landau level, the Bohr magneton and the g-factor
of the electron. The partial densities of spin up or spin down electrons and
the corresponding electron spin asymmetry density are

$\rho_{e\sigma}=\frac{eB}{(2\pi)^{2}}\sum_{N_{e}}\int_{-\infty}^{\infty}dp_{z}f_{e\sigma}(\varepsilon_{e\sigma}(p_{z},N_{e}),T)$
(26)

and

$W_{e}=\rho_{e\uparrow}-\rho_{e\downarrow}\ ,$ (27)

respectively. Note that, as in the case of protons, in Eqs. (24), (26) and
(27) the sum over the electron Landau levels $N_{e}$ runs from 0 up to a
maximum level obtained numerically as in the case of the maximum proton Landau
level. For details, the reader is again referred to the appendix B of Ref.
Bauer20 .

The last remained element to be defined is the quantity $L_{p}$ (see Eq. (9))
of the explicit magnetic field contribution to the total energy density. This
quantity is simply

$L_{p}=\frac{eB}{(2\pi)^{2}}\sum_{N_{p}}N_{p}\sum_{\sigma=\uparrow,\downarrow}\int_{-\infty}^{\infty}dp_{z}f_{p\sigma}(\varepsilon_{p\sigma}(p_{z},N_{p}),T)\
.$ (28)

Once we have the total energy density, the Helmhotz total free energy density,
from which the chemical potentials of all the particle species can be
evaluated, is easily obtained from the usual thermodynamical relation

${\cal F}={\cal E}-T{\cal S}\ ,$ (29)

where ${\cal S}$ is total entropy density

${\cal S}={\cal S}_{n}+{\cal S}_{p}+{\cal S}_{e}\ ,$ (30)

with ${\cal S}_{n}$, ${\cal S}_{p}$ and ${\cal S}_{e}$ the corresponding
neutron, proton and electron contributions:

$\displaystyle{\cal S}_{n}$ $\displaystyle=$
$\displaystyle-\sum_{\sigma=\uparrow,\downarrow}\frac{1}{(2\pi)^{3}}\int
d^{3}\vec{p}\left[f_{n\sigma}\mbox{ln}(f_{n\sigma})+(1-f_{n\sigma})\mbox{ln}(1-f_{n\sigma})\right]\
,$ $\displaystyle{\cal S}_{p}$ $\displaystyle=$
$\displaystyle-\sum_{\sigma=\uparrow,\downarrow}\frac{eB}{(2\pi)^{2}}\sum_{N_{p}}\int_{-\infty}^{\infty}dp_{z}\left[f_{p\sigma}\mbox{ln}(f_{p\sigma})+(1-f_{p\sigma})\mbox{ln}(1-f_{p\sigma})\right]\
,$ $\displaystyle{\cal S}_{e}$ $\displaystyle=$
$\displaystyle-\sum_{\sigma=\uparrow,\downarrow}\frac{eB}{(2\pi)^{2}}\sum_{N_{e}}\int_{-\infty}^{\infty}dp_{z}\left[f_{e\sigma}\mbox{ln}(f_{e\sigma})+(1-f_{e\sigma})\mbox{ln}(1-f_{e\sigma})\right]\
,$ (31)

where we have omitted the explicit dependencies of the Fermi–Dirac
distributions to simplify the notation.

Once ${\cal F}$ is known one can obtain the pressure of the system simply as

$P=\rho\left(\frac{\partial{\cal F}}{\partial\rho}\right)_{T,B}-{\cal F}\ ,$
(32)

from which is possible to determine the isothermal compressibility

${\cal K}=\left[\rho\left(\frac{\partial
P}{\partial\rho}\right)_{T,B}\right]^{-1}\ .$ (33)

##  III Results and Discussion

In the following we discuss the properties of hot and dense neutron star
matter under the presence of strong magnetic fields. Results are presented for
densities up to 0.4 fm-3, temperatures $T=5$, $15$ and $30$ MeV, and the
magnetic fields strengths $B=10^{16}$, $10^{17}$ and $10^{18}$ G.

Figure 1: (Color online) Particle fractions $x_{i}\equiv\rho_{i}/\rho$
($i=n_{\uparrow},n_{\downarrow},p_{\uparrow},p_{\downarrow},e_{\uparrow},e_{\downarrow}$)
of $\beta$-stable matter at $T=5$ MeV for the magnetic field strengths,
$10^{16},10^{17}$ and $10^{18}$ G for the two interactions considered, LNS
(panels (a), (c) and (e)) and BSk21 (panels (b), (d) and (f)).

We start by showing in Fig. 1 the fractions of neutrons, protons and electrons
with spin up and down ($x_{i}\equiv\rho_{i}/\rho$,
$i=n_{\uparrow},n_{\downarrow},p_{\uparrow},p_{\downarrow},e_{\uparrow},e_{\downarrow}$)
in $\beta$-stable matter, obtained by solving Eqs. (4), (5) and (7), at $T=5$
MeV for the three magnetic field strengths just mentioned and for the two
interactions considered, LNS (panels (a), (c) and (e)), and BSk21 (panels (b),
(d) and (f)). As it is seem in the figure, a magnetic field of strength
$10^{16}$ G induces only a extremely low polarization of the spins of the
different components of neutron star matter at very low densities, and fields
of the order of at least $10^{17}$ G are needed to see appreciable differences
in the fractions of neutrons, protons and electrons with opposite spin
projections. We note also that whereas the fraction of protons with spin up
(i.e., oriented parallel to the magnetic field) is larger than the fraction of
protons with spin down the opposite is observed in the case of neutrons and
electrons. This is simply a consequence of the fact that the proton g-factor
is positive while the neutron and electron ones are negative. Due to this,
spin up (down) protons (neutrons and electrons) have lower energy than spin
down (up) protons (neutrons and electrons) (see Eqs. (21), (22) and (25)).
Consequently, the configurations of the system with fractions of spin up
protons larger than spin down protons, and fractions of spin down neutrons and
electrons larger than spin up neutrons and electrons, have less energy and,
therefore, are physically favorable. We observe that although in the case of
the BSk21 interaction, for each particle species $i$, the difference in the
fractions between the spin up and spin down component decreases with
increasing density, in the case of the LNS an increase of this difference is
observed for neutrons and protons for densities $\rho\gtrsim 0.3$ fm-3. This
is a consequence of the appearance of a ferromagnetic instability predicted by
the LNS model at high densities, instability that is corrected in the case of
the BSk21 force as we mentioned before.

Figure 2: (Color online) Spin asymmetry $W_{j}/\rho_{j}$ of each particle
species for neutron star matter at several temperatures and magnetic field
strengths for the LNS interaction. Figure 3: (Color online) Spin asymmetry
$W_{j}/\rho_{j}$ of each particle species for neutron star matter at several
temperatures and magnetic field strengths for the BSk21 interaction. Figure 4:
(Color online) Neutron and proton effective masses predicted by the LNS (thick
lines) and the BSk21 (thin lines) interactions as function of the density for
the three temperatures considered in this work.

To understand better the state of spin polarization of the system, we show now
the spin asymmetry, defined as the ratio
$W_{j}/\rho_{j}=(\rho_{j\uparrow}-\rho_{j\downarrow})/\rho_{j}$ with
$\rho_{j}=\rho_{j\uparrow}+\rho_{j\downarrow}$ ($j=n,p,e$), of each particle
species for the three temperatures and the three magnetic fields considered in
this work. Results for the LNS and the BSk21 models are presented in Fig. 2
and Fig. 3, respectively. We note first that the value $W_{j}/\rho_{j}=0$
corresponds to the case in which the species $j$ is unpolarized, whereas
$W_{j}/\rho_{j}=\pm 1$ means that this species is totally polarized, i.e., all
its spins are aligned along the same direction, parallel ($W_{j}/\rho_{j}=1$)
or antiparallel ($W_{j}/\rho_{j}=-1$) to the one defined by the magnetic
field. Partially polarized configurations of a species $j$ correspond to
values of $W_{j}/\rho_{j}$ between $-1$ and $1$. To begin with, we observe in
both figures that a magnetic field of $10^{16}$ G has almost no effect on the
spin asymmetry of the different particles, being the system essentially in a
global unpolarized state. Just for the lowest temperature $T=5$ MeV and at
very low densities this field induces a very tiny polarization of the particle
spins, especially on the electron ones. Only magnetic fields with a strength
$B\geq 10^{17}$ G are able to change the spin polarization of the system from
the unpolarized state to a partially polarized one. As the sign of the spin
asymmetry of each particle indicates, while protons have their majority of
their spins oriented parallel to the magnetic field ($W_{p}/\rho_{p}>0$),
neutron and electron spins are mostly aligned in the opposite direction
($W_{n}/\rho_{n}<0$, $W_{e}/\rho_{e}<0$). This, as it was discussed above, is
because system configurations with larger fractions of spin up protons and
spin down neutrons and electrons are energetically favored over those with
larger fractions of spin down protons and spin up neutrons and electrons. We
note that although the spin asymmetry of the protons (neutrons) decreases
(increases) always with density for the BSk21 interaction, indicating the
opposition of this nuclear interaction to the spin polarization induced by the
magnetic field, this is not the case when the LNS one is used. In this case,
we observe that $W_{p}/\rho_{p}$ ($W_{n}/\rho_{n}$) decreases (increases) of
up to a density $\rho\sim 0.2$ fm-3 for $B=10^{17}$ G ($\rho\sim 0.3$ fm-3 for
$B=10^{18}$ G) and then it increases (decreases). This behavior is again just
a consequence of the ferromagnetic instability predicted by the LNS force.
Regarding the electrons, we see that their spin asymmetry always increases
monotonously with density, reaching asymptotically their unpolarized state
($W_{e}/\rho_{e}=0$) at high densities. We observe also that protons and
electrons are more polarized than neutrons in all the range of densities
explored, being the lower degree of polarization of the neutrons due to its
weak anomalous magnetic moment. Note finally, that the spin asymmetry of the
three species decreases (in absolute value) when increasing the temperature.
This is expected since increasing temperature increases the entropy of the
system and, consequently, its disorder. The number of spin up and spin down
particles becomes more and more similar and, therefore, the system and becomes
less polarized.

In Fig. 4 we show now the effect of the magnetic field on the neutron and
proton effective masses predicted by the LNS (thick lines) and the BSk21 (thin
lines) interactions for the three temperatures considered in this work. First,
we observe that a magnetic field of $10^{16}$ G is too low to have any effect
on the effective mass of both neutrons and protons with different spin
projections because, as shown in panels (a), (d) and (g) of Figs. 2 and 3, it
has almost no effect on their spin asymmetry and, consequently, their
effective masses are essentially spin independent for this field. Second, we
note that the increase of $m^{*}_{\tau\sigma}/m_{\tau}$ above one seen in the
case of the BSk21 interaction is a consequence of the density dependence of
the effective mass predicted by this force which goes as
$(a+b\rho+c\rho^{1+\beta}+d\rho^{1+\gamma})^{-1}$ (see Eq. (14)), whereas the
LNS interaction, for which the coefficients $c_{0},\cdot\cdot\cdot,c_{3}$ are
zero, predicts $m^{*}_{\tau\sigma}\sim(\tilde{a}+\tilde{b}\rho)^{-1}$.
Finally, we notice for both interactions that the effective mass of neutrons
and protons is in general larger for the more abundant of their spin
projection component, respectively, spin down neutrons and spin up protons.
Note, however, that the BSk21 interaction in the case of protons predicts
$m^{*}_{p_{\downarrow}}>m^{*}_{p_{\uparrow}}$ for densities below $\sim 0.1$
fm-3. The splitting of the spin up and spin down nucleon effective masses can
be understood by looking at the term
$\frac{s_{\sigma}}{4}\sum_{\tau^{\prime}=n,p}(b_{1}+c_{1}+(b_{3}+c_{3})I_{\tau}I_{\tau^{\prime}})W_{\tau^{\prime}}$
of Eq. (14). To facilitate the analysis of the role of this term on the
effective masses, in Fig. 5 we show its density dependence for the two
interactions considered for neutron star matter at $T=5$ MeV in the presence
of a magnetic field of $10^{18}$ G. Let us consider first the case of the LNS
interaction. As it is seen in panel (a), for this model this term is always
positive for spin up neutrons and spin down protons while it is negative for
spin down neutrons and spin up protons. Therefore, is it clear from the
definition of the effective mass given in Eq. (14) that, for the LNS
interaction, the contribution of this term leads to values of
$m^{*}_{n_{\downarrow}}$ and $m^{*}_{p_{\uparrow}}$ always larger than those
of $m^{*}_{n_{\uparrow}}$ and $m^{*}_{p_{\downarrow}}$, respectively. For the
BSk21 interaction (panel (b)) we observe that also in this case this term is
always positive for spin up neutrons and negative for spin down neutrons and,
therefore, $m^{*}_{n_{\downarrow}}>m^{*}_{n_{\uparrow}}$ in the whole range of
densities explored. On the other hand, for densities below $\sim 0.1$ fm-3,
this term is positive (negative) for spin up (down) protons and vice versa for
densities above this value. Therefore, as it is seen in Fig. 4, this
interaction predicts $m^{*}_{p_{\downarrow}}>m^{*}_{p\uparrow}$ for $\rho<0.1$
fm-3 and $m^{*}_{p_{\uparrow}}>m^{*}_{p_{\downarrow}}$ for $\rho>0.1$ fm-3.
Similar conclusions can be drawn from the analysis of results obtained for
other temperatures and magnetic fields.

Figure 5: (Color online) Density dependence of the term
$\frac{s_{\sigma}}{4}\sum_{\tau^{\prime}=n,p}(b_{1}+c_{1}+(b_{3}+c_{3})I_{\tau}I_{\tau^{\prime}})W_{\tau^{\prime}}$
of Eq. (14) for the LNS (panel (a)) and BSk21 (panel (b)) interactions for
neutron star matter at $T=5$ MeV in the presence of a magnetic field of
$10^{18}$ G.

Let us finish this section by analyzing the bulk thermodynamical properties of
the system. We show first in Fig. 6 the Helmhotz total free energy density as
a function of the density for the two interactions and the different
temperatures and magnetic fields considered in this work. As it can be seen
the effect of the magnetic field seems to be almost negligible. The reason is
mainly due to the low value of the nuclear magneton which makes the
interaction of neutrons and protons with the magnetic field too mild for the
values of $B$ considered. Consequently, the contribution to the energy density
from the interaction of nucleons with the field (last term of Eq. (9)) is too
small, and that from the nuclear interaction ${\cal E}_{nuc}$ (Eq. (10))
depends also very little on it. In addition, also the electron contribution
${\cal E}_{elec}$ (Eq. (24)) to the total energy density depends very weakly
on the field, increasing slightly when increasing $B$. The reason, in this
case, should not be attributed to the value of the Bohr magneton, three orders
of magnitude larger than the nuclear one, but rather to the fact that the
total electron fraction is very low (see Fig. 1). For illustration we show,
for the two model interactions considered, in Tabs. 1 (LNS) and 2 (BSk21) all
the contributions to the Helmholtz total free energy density for three
representative densities $\rho=0.08$ fm-3, $\rho=0.16$ fm-3 and $\rho=0.32$
fm-3, a temperature $T$ of 5 MeV, and the magnetic fields $B=10^{16}$ G and
$B=10^{18}$ G. Note that also the neutron, proton and electron contributions
to the total entropy density depend very little on the magnetic field.

Finally, in Figs. 7 and 8 we show, respectively, the pressure and the
isothermal compressibility of the system as a function of the density for the
two interactions and the different temperatures and magnetic fields considered
in this work. The pressure, as it is required by the stability conditions,
increases monotonically with the density and is larger for larger values of
the temperature. Note that, as expected from our previous analysis of the
Helmhotz total free energy density, both the pressure and the isothermal
compressibility present also a very mild dependence on the magnetic field
which is almost imperceptible in the figures. As it is seen, the isothermal
compressibility decreases monotonously with density from relatively high
values at low densities, and it becomes very small for $\rho\gtrsim 0.3$ fm-3,
showing that from this density on (for these two particular interactions)
highly magnetized neutron star matter can be considered an almost
incompressible system.

Figure 6: (Color online) Helmhotz total free energy density as a function of the density for the two interactions and the different temperatures and magnetic fields considered in this work. Figure 7: (Color online) Pressure of the system as a function of the density for the two interactions and the different temperatures and magnetic fields considered in this work. Figure 8: (Color online) Isothermal compressibility as a function of the density for the two interactions and the different temperatures and magnetic fields considered in this work. $B$ | $\rho$ | ${\cal E}_{nucl}$ | ${\cal E}_{elec}$ | $\mu_{N}B\left(2L_{p}+\rho_{p}-\frac{g_{p}}{2}W_{p}+\frac{g_{n}}{2}W_{n}\right)$ | ${\cal S}_{p}T$ | ${\cal S}_{n}T$ | ${\cal S}_{e}T$ | ${\cal F}$
---|---|---|---|---|---|---|---|---
| $0.08$ | $0.625$ | $0.088$ | $0.014$ | $0.042$ | $0.253$ | $0.008$ | $0.424$
$10^{16}$ | $0.16$ | $1.836$ | $0.402$ | $0.056$ | $0.086$ | $0.310$ | $0.016$ | $1.882$
| $0.32$ | $8.264$ | $1.630$ | $0.237$ | $0.141$ | $0.365$ | $0.033$ | $9.592$
| $0.08$ | $0.606$ | $0.144$ | $-0.064$ | $0.043$ | $0.251$ | $0.004$ | $0.388$
$10^{18}$ | $0.16$ | $1.783$ | $0.563$ | $-0.084$ | $0.079$ | $0.310$ | $0.018$ | $1.855$
| $0.32$ | $8.259$ | $1.939$ | $-0.108$ | $0.123$ | $0.367$ | $0.031$ | $9.569$
Table 1: Separate contributions to the Helmhotz total free energy density for densities $\rho=0.08$ fm-3, $\rho=0.16$ fm-3 and $\rho=0.32$ fm-3, a temperature $T=5$ MeV, and the magnetic fields $B=10^{16}$ G and $B=10^{18}$ G for the LNS interaction. The Helmhotz total free energy density is shown in the last column. Units are given in MeV fm-3. $B$ | $\rho$ | ${\cal E}_{nucl}$ | ${\cal E}_{elec}$ | $\mu_{N}B\left(2L_{p}+\rho_{p}-\frac{g_{p}}{2}W_{p}+\frac{g_{n}}{2}W_{n}\right)$ | ${\cal S}_{p}T$ | ${\cal S}_{n}T$ | ${\cal S}_{e}T$ | ${\cal F}$
---|---|---|---|---|---|---|---|---
| $0.08$ | $0.612$ | $0.104$ | $0.017$ | $0.046$ | $0.272$ | $0.008$ | $0.407$
$10^{16}$ | $0.16$ | $1.590$ | $0.362$ | $0.050$ | $0.082$ | $0.296$ | $0.016$ | $1.608$
| $0.32$ | $9.700$ | $1.720$ | $0.257$ | $0.139$ | $0.245$ | $0.035$ | $11.258$
| $0.08$ | $0.635$ | $0.151$ | $-0.100$ | $0.046$ | $0.271$ | $0.004$ | $0.365$
$10^{18}$ | $0.16$ | $1.623$ | $0.489$ | $-0.140$ | $0.079$ | $0.297$ | $0.019$ | $1.577$
| $0.32$ | $9.880$ | $1.964$ | $-0.087$ | $0.134$ | $0.247$ | $0.032$ | $11.344$
Table 2: As Tab. 1 for the BSk21 interaction.

##  IV Summary and conclusions

In the present work we have studied the properties of hot and dense neutron
star matter under the presence of strong magnetic fields using two Skyrme
interactions, namely the LNS interaction developed by Cao et al., Cao06  and
the interaction BSk21 Goriely10  of the Brussels–Montreal group. In
particular, we have constructed the equation of state of the system and
analyze its composition for a range of densities, temperatures and magnetic
field intensities of interest for the study of supernova and proto-neutron
star matter, with a particular interest on the degree of spin-polarization of
the different components. Our results show that in order to see appreciable
differences in the fractions of neutrons, protons and electrons with opposite
spin projections the intensity of the magnetic field should be at least of the
order of $10^{17}$ G. They also show that system configurations with larger
fractions of spin up protons and spin down neutrons and electrons are
energetically favored over those with larger fractions of spin down protons
and spin up neutrons and electrons. We have also studied the effect of the
magnetic field on the neutron and proton effective masses finding that, for
the two interactions considered, that the effective mass of neutrons and
protons is in general larger for the more abundant of their spin projection
component, respectively, spin down neutrons and spin up protons. Finally, we
have determined the bulk thermodynamical properties of the system, finding
that the effect of the magnetic field on the Helmhotz total free energy
density, pressure and isothermal compressibility of the system is almost
negligible due to the low value of the nuclear magneton which makes the
interaction of neutrons and protons with the magnetic field too mild for the
values of B considered.

## Acknowledgements

I.V. thanks the support of the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No. 824093.

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