# Landau quantization and spin polarization of cold magnetized quark matter

Zhen-Yan Lu<EMAIL_ADDRESS>Hunan Provincial Key Laboratory of
Intelligent Sensors and Advanced Sensor Materials,
School of Physics and Electronics, Hunan University of Science and Technology,
Xiangtan 411201, China  Jian-Feng Xu<EMAIL_ADDRESS>School of Physics and
Electrical Engineering, Anyang Normal University, Anyang 455000, China  Xin-
Jian Wen<EMAIL_ADDRESS>Department of Physics and Institute of Theoretical
Physics, Shanxi University,
Taiyuan 030006, China  Guang-Xiong Peng<EMAIL_ADDRESS>School of Nuclear
Science and Technology, University of Chinese Academy of Sciences,
Beijing 100049, China  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Marco Ruggieri<EMAIL_ADDRESS>School of
Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

###### Abstract

The magnetic field and density behaviors of various thermodynamic quantities
of strange quark matter under compact star conditions are investigated in the
framework of the thermodynamically self-consistent quasiparticle model. For
individual species, a larger number density $n_{i}$ leads to a larger magnetic
field strength threshold that align all particles parallel or antiparallel to
the magnetic field. Accordingly, in contrast to the finite baryon density
effect which reduces the spin polarization of magnetized strange quark matter,
the magnetic field effect leads to an enhancement of it. We also compute the
sound velocity as a function of the baryon density and find the sound velocity
shows an obvious oscillation with increasing density. Except for the
oscillation, similar to the zero-magnetic field case that the sound velocity
grows with increasing density and approaches the conformal limit
$V_{s}^{2}=1/3$ at high densities from below.

##  I Introduction

Quarks are generally believed to appear within hadronic matter, due to the
feature of color confinement. However, it is expected, with increasing
density, that the basic constituent of hadronic matter, i.e. hadrons, might
squeeze out to form deconfined quark matter at low temperatures. About forty
years ago, Witten first conjectured that strange quark matter (SQM) Farhi and
Jaffe (1984), consisting of roughly equal number of up, down, and strange
quarks, could be the true ground state of strong interaction Witten (1984).

Since SQM could be absolutely stable, there might be compact objects with a
quark core or even completely made of quarks and leptons, i.e., the so-called
strange stars Haensel, Zdunik, and Schaeffer (1986); Olinto (1987); Alcock,
Farhi, and Olinto (1986); Xu, Zhang, and Qiao (2001); Weber (2005). Recently,
comparison of the theoretical calculations in a model-independent way with
astrophysical observations suggests that there might be a quark matter core in
the maximally massive neutron stars Annala  _et al._ (2020). Several studies
also imply that strange stars can coexist with neutron stars Bombaci and Datta
(2000); Drago  _et al._ (2016); Bhattacharyya  _et al._ (2017); Drago,
Lavagno, and Pagliara (2014); Drago and Pagliara (2016); Bombaci  _et al._
(2021); Drago and Pagliara (2020); Lourenço  _et al._ (2021). One important
feature of such stars is the associated strong magnetic field, with strength
of the order $10^{11}\sim 10^{13}$ G on the surface of pulsars Haensel,
Potekhin, and Yakovlev (2007), or even reaches $10^{14}\sim 10^{15}$ G in some
magnetars Duncan and Thompson (1992). In the core of compact stars, the
magnetic field strength is estimated as high as $10^{18}\sim 10^{20}$ G Lai
and Shapiro (1991); Bocquet  _et al._ (1995); Broderick, Prakash, and Lattimer
(2002); Ferrer  _et al._ (2010); Ferrer and de la Incera (2013). Except for
the stellar objects, the magnetic field strength produced in the heavy ion
collisions can be as strong as $10^{18}\sim 10^{19}$ G Kharzeev, McLerran, and
Warringa (2008); Fukushima, Kharzeev, and Warringa (2008); Skokov, Illarionov,
and Toneev (2009); Voronyuk  _et al._ (2011); Deng and Huang (2012). However,
let us emphasize that estimates of the largest value of magnetic field $B_{m}$
inside the compact stars exist, see for example Ref. Chatterjee  _et al._
(2015) and references therein; however, these estimates have some dependence
on the model used for the equation of state of bulk matter. In this study we
take the $B_{m}=O(10^{18}~{}\mathrm{G})$ as a conservative estimate of the
maximum value of $B_{m}$, and use larger values of $B_{m}$ only for
illustrative purposes, see Fig. 5.

The presence of a strong magnetic field introduces Landau levels for the
charged particles and hence has significant effects on the stability of SQM
Chakrabarty (1996); Isayev (2015); Menezes  _et al._ (2009a). The properties
of SQM in a strong magnetic field have attracted increasing interest over the
past few decades, for example, the effects of an external magnetic field on
the symmetry energy Chu  _et al._ (2018, 2020), equation of state Isayev
(2018) and surface tension of quark matter Lugones and Grunfeld (2019), and
$r$-mode instability Huang  _et al._ (2010); Xu  _et al._ (2021) as well as
structure properties of strange stars Hou  _et al._ (2015); Peres Menezes and
Laércio Lopes (2016); Chu, Chen, and Wang (2014); Kayanikhoo, Naficy, and
Bordbar (2020); Chu  _et al._ (2021), etc. In addition, the presence of a
magnetic field can also affect the in-medium chiral condensates Menezes  _et
al._ (2009b), and hence the critical temperature of the chiral phase
transition Ruggieri, Lu, and Peng (2016); Wen, He, and Liu (2021); Ferrer and
Hackebill (2020); Xu, Chao, and Huang (2021); Chaudhuri  _et al._ (2019,
2020); Grunfeld  _et al._ (2014); Farias  _et al._ (2021), as well as the
deconfinement transition Fraga and Palhares (2012); Mizher, Chernodub, and
Fraga (2010); Ferreira  _et al._ (2014); Gatto and Ruggieri (2011); Backes,
Marquezb, and Menezes (2021). Due to the intractable nature of quantum
chromodynamics (QCD) in the nonperturbative regime, we need to find a proper
way to mimic the strong interactions between quarks. The medium-dependence of
quarks can be taken into account by considering a chemical potential and/or
temperature dependent quark mass in the quasiparticle model Peshier  _et al._
(1994). In this case, however, special attention should be paid to the
thermodynamic consistency of the phenomenological models Gorenstein and Yang
(1995); Bannur (2007); Alba  _et al._ (2014); Xu  _et al._ (2015a, b); Xia
_et al._ (2014). To confine quarks, an effective bag constant is required in
the nonperturbative regime111Recently, there was a discussion on the quark
mean-field model for nuclear matter with or without a bag constant and quark
confinement is found to be mainly demonstrated by the bag after it is included
in the model, instead of the confining potential Zhu  _et al._ (2019).. Since
the quark masses are chemical potential and/or temperature dependent, the
additional effective bag constant should also be simultaneously chemical
potential and/or temperature dependent in order to satisfy the fundamental
relation of thermodynamics Peshier, Kampfer, and Soff (2000); Ma  _et al._
(2019). Following the original idea in Refs. Schertler, Greiner, and Thoma
(1997); Schertler  _et al._ (1998), the quasiparticle model with a fixed
strong coupling has been used to study the finite-size strangelets Wen  _et
al._ (2009) and the medium effects on the surface tension of strangelets Wen
_et al._ (2010); Wen, Yang, and Su (2011). Later, this model is extended to
include the running strong coupling with a chemical potential dependent
renormalization subtraction point, which is constrained by the Cauchy
condition in the chemical potential space Lu  _et al._ (2016a).

In this work, we restrict ourselves to the deconfined SQM and study the effect
of a magnetic field on the properties of SQM, especially on the behavior of
maximum Landau levels, particle fractions, relative spin polarization, and
sound velocity at finite baryon densities in the presence of a uniform
external magnetic field. We can anticipate the main new results of our study:
within the quasiparticle model, we study for the first time the polarization
of dense strange quark matter in a strong magnetic background field, and we
complete this investigation by the calculation of the squared speed of sound.
The paper is outlined as follows. In Sec. II, we review the thermodynamic
consistent quasiparticle model with a chemical potential dependent bag
constant in strong magnetic fields. In Sec. III, we present the numerical
results and give discussions for our calculations. Finally, a summary is given
in Sec. IV.

##  II Thermodynamic consistent quasiparticle model

Unlike the density-dependent mass model that the quark masses are density-
dependent Chakrabarty (1991); Peng  _et al._ (2000); Wen  _et al._ (2005); Lu
_et al._ (2016b); Chu and Chen (2014), in the quasiparticle model, the strong
interactions between quarks are mimicked by arranging a chemical potential
dependence of quark masses. For the medium dependence, the effective quark
mass used in this work was derived in the zero momentum limit of the
dispersion relations following from an effective quark propagator Schertler,
Greiner, and Thoma (1997) by resuming one-loop self-energy diagrams in the
hard dense loop approximation Vija and Thoma (1995); Weldon (1982)

$m_{i}^{*}=\frac{m_{i0}}{2}+\sqrt{\frac{m_{i0}^{2}}{4}+\frac{g^{2}\mu_{i}^{2}}{6\pi^{2}}},$
(1)

where $m_{i0}$ and $\mu_{i}$ are respectively the current quark mass and
chemical potential of quark flavor $i$.

In Refs. Wen  _et al._ (2009); Zhang  _et al._ (2021), the properties of
strange quark matter with and without finite size effects are investigated in
the quasiparticle model, in which the strong coupling is treated as a pure
constant. However, it is well-known that the strong coupling $g$ is running
with the energy scale, and here we adopt the following phenomenological
expression Patra and Singh (1996)

$g^{2}=\frac{48\pi^{2}}{29\ln\left(a\mu_{i}^{2}/\Lambda^{2}\right)},$ (2)

where $a=0.8$, and $\Lambda$ is the QCD scale parameter controlling the rate
at which QCD coupling runs as a function of energy scale. In the vanishing
current mass limit, Eq. (1) reduces to

$m_{i}^{*}=\frac{g\mu_{i}}{\sqrt{6}\pi},$ (3)

which can be used as the effective masses for up and down quarks since their
current masses are small compared to the strange quark. Note that since
electrons do not participate in the strong interactions, their masses keep
constant as in the normal case.

To consider the effects of a strong magnetic field on the Landau quantization
and the spin polarization of the strongly interacting quark matter, we need to
know the energy spectrum of the spin-half particle in a strong magnetic field.
Without loss of generality, we assume the external magnetic field along with
the $z$-direction, i.e., $\bm{\mathcal{B}}=B_{m}\hat{z}$. In consequence, the
energy spectrum of the spin-half charged particle can be obtained by solving
the Dirac equation. We accordingly have

$\displaystyle\varepsilon_{i}=\sqrt{p_{z}^{2}+\bar{m}_{i,\nu}^{*2}}.$ (4)

Here $\bar{m}_{i,\nu}^{*}=\sqrt{m_{i}^{*2}+2\nu|q_{i}|B_{m}}$ is the effective
mass of particle $i$ in the presence of an external magnetic field, $p_{z}$ is
the particle momentum in the $z$-direction, $q_{i}$ is the electronic charge
of particle $i$, and the Landau level $\nu$ is defined as Broderick, Prakash,
and Lattimer (2000); Strickland, Dexheimer, and Menezes (2012)

$\displaystyle\nu=l+\frac{1}{2}-\frac{\eta}{2}\frac{q_{i}}{|q_{i}|},$ (5)

where $l$ denotes the orbital angular momentum, and $\eta=\pm 1$ represents
the two eigenstates of the spin-half charged particle with “+1” for spin up
and “-1” for spin down. Due to the Landau quantization, the integral over the
momentum components perpendicular to the magnetic field become discrete.
Consequently, we have

$\int\int\int dp_{x}dp_{y}dp_{z}\rightarrow 2\pi|q_{i}|B_{m}\sum_{\eta=\pm
1}\sum_{l}\int\mathrm{d}p_{z}.$ (6)

Rewrite Eq. (4) in terms of the chemical potential, and define the maximum
$p_{z}$ as $p_{z,F}=\sqrt{\mu_{i}^{2}-\bar{m}_{i,\nu}^{*2}}$, we then have an
upper limit for the Landau levels

$\nu\leq\nu_{i,\max}=\text{Int}\left[\frac{\mu_{i}^{2}-m_{i}^{*2}}{2|q_{i}|B_{m}}\right]$
(7)

due to the fact that the Fermi momenta $p_{z,F}$ must be real-valued
quantities. In Eq. (7), the symbol $\text{Int}[...]$ represents floor of the
enclosed quantity.

At zero temperature, the quasiparticle contribution
$\Omega_{i}\equiv\Omega_{i}(\mu_{i},m_{i}^{*})$ to the total thermodynamic
potential density for magnetized quark matter is given by

$\displaystyle\Omega_{i}$ $\displaystyle=$
$\displaystyle-\frac{d_{i}|q_{i}|B_{m}}{4\pi^{2}}\sum_{\eta=\pm
1}\sum_{l}\Bigg{\\{}\mu_{i}\sqrt{\mu_{i}^{2}-\bar{m}_{i,\nu}^{*2}}$ (8)
$\displaystyle-\bar{m}_{i,\nu}^{*2}\ln\Bigg{(}\frac{\mu_{i}+\sqrt{\mu_{i}^{2}-\bar{m}_{i,\nu}^{*2}}}{\bar{m}_{i,\nu}^{*}}\Bigg{)}\Bigg{\\}},$

with the degenerate factor $d_{i}=1$ for electron and $d_{i}=3$ for quarks
respectively. The pressure and energy density for magnetized SQM within the
quasiparticle model are given by

$\displaystyle P$ $\displaystyle=$ $\displaystyle-\Omega-B^{*},$ (9)
$\displaystyle E$ $\displaystyle=$
$\displaystyle\Omega+\sum_{i}\mu_{i}n_{i}+B^{*},$ (10)

where $\Omega\equiv\sum_{i}\Omega_{i}$ is the total thermodynamic potential
density222A term $B_{m}^{2}/2$ coming from the magnetic field contribution has
been dropped since it is irrelevant for the present work. containing a
summation of all quasiparticle contributions in Eq. (8), while
$B^{*}\equiv\sum_{i}B_{i}(\mu_{i},m_{i}^{*})+B_{0}$ is the chemical potential
dependence of effective bag constant introduced to fulfill the thermodynamic
self-consistency requirement Wen  _et al._ (2009). $B_{0}$ is the MIT bag
constant, which is taken to be zero since it is not numerically relevant to
our calculations. Note that there is no need to introduce an effective bag
constant for electrons since they do not participate in the strong
interactions and thus do not have a chemical potential dependent mass. For a
quasiparticle Fermi system the number density of the component $i$ has the
same form of the free-particle case in the presence of an external magnetic
field, which is given by

$\displaystyle n_{i}=\frac{d_{i}|q_{i}|B_{m}}{2\pi^{2}}\sum_{\eta=\pm
1}\sum_{l}\sqrt{\mu_{i}^{2}-\bar{m}_{i,\nu}^{*2}}.$ (11)

According to the fundamental differential equation of thermodynamics, the
number density $n_{i}$ is obtained by taking the first derivative of the
thermodynamic potential density with respect to the corresponding chemical
potential $\mu_{i}$. Mathematically, this is equivalent to requiring

$\displaystyle
n_{i}=-\left.\frac{\mathrm{d}\Omega}{\mathrm{d}\mu_{i}}\right|_{\mu_{j\neq
i}}$ $\displaystyle=$
$\displaystyle-\frac{\partial\Omega_{i}}{\partial\mu_{i}}-\underbrace{\left[\frac{\partial\Omega_{i}}{\partial
m_{i}^{*}}\frac{\mathrm{d}m_{i}^{*}}{\mathrm{d}\mu_{i}}+\frac{\partial
B^{*}}{\partial\mu_{i}}\right]}_{=0}.~{}~{}$ (12)

In the above, the first term on the right hand side of the second equality is
in fact equal to Eq. (11), while the vanishing of second term is required to
satisfy the thermodynamic consistency requirement of the quasiparticle model
Lu  _et al._ (2016a). From Eq. (12), we have

$\displaystyle\frac{\mathrm{d}B^{*}}{\mathrm{d}\mu_{i}}\frac{\mathrm{d}\mu_{i}}{\mathrm{~{}d}m_{i}^{*}}=-\frac{\partial\Omega_{i}}{\partial
m_{i}^{*}}$ (13)

or equivalently

$\displaystyle B^{*}$ $\displaystyle=$
$\displaystyle-\frac{d_{i}|q_{i}|B_{m}}{2\pi^{2}}\sum_{\eta=\pm
1}\sum_{l}\int_{\mu_{i}^{c}}^{\mu_{i}}m_{i}^{*}\frac{\mathrm{d}m_{i}^{*}}{\mathrm{d}\mu_{i}}$
(14)
$\displaystyle\times\ln\Bigg{(}\frac{\mu_{i}+\sqrt{\mu_{i}^{2}-\bar{m}_{i,\nu}^{*2}}}{\bar{m}_{i,\nu}^{*}}\Bigg{)}\mathrm{d}\mu_{i}.$

To ensure the positive of square root in Eq. (14), the lower limit of the
integration over $\mu_{i}$ in Eq. (14) should satisfy

$\mu_{i}^{c2}-\bar{m}_{i,\nu}^{*2}\geq 0.$ (15)

By using the effective quark mass in Eq. (1), the medium dependent effective
bag constant $B^{*}$ can be derived by numerical integration.

##  III Numerical Results and discussions

In this section, we will present our results on the properties of various
thermodynamic quantities in the presence of an external strong magnetic field
at nonzero baryon densities. For the stable SQM, the $\beta$-equilibrium can
be reached by the weak reactions $d,s\leftrightarrow u+e+\bar{v}_{e}$ and
$\quad s+u\leftrightarrow u+d$. Correspondingly, we have the following
conditions for the relevant chemical potentials as

$\displaystyle\begin{cases}\mu_{d}=\mu_{s}\equiv\mu,\cr\mu_{u}+\mu_{e}=\mu.\end{cases}$
(16)

Here the chemical potential of neutrinos is set to zero because they can enter
or leave the system freely. We also have the expressions of the baryon density

$\displaystyle n_{b}=\frac{1}{3}(n_{u}+n_{d}+n_{s}),$ (17)

and the charge neutrality condition

$\displaystyle\frac{2}{3}n_{u}-\frac{1}{3}n_{d}-\frac{1}{3}n_{s}-n_{e}=0,$
(18)

which has to be fulfilled for stable SQM presented in compact stars.

###  III.1 Chemical potentials and Landau levels

Figure 1: (Color online) Quark and electron chemical potentials as functions
of the baryon density for two different magnetic field strengths.
Figure 2: (Color online) The maximum Landau levels for up, down, strange
quarks, and electron change with the baryon density.

Strange stars are hypothetical objects consisting of stable SQM in
$\beta$-equilibrium condition. For a given baryon density, one can numerically
solve the set of equations in (16), (17), and (18) to obtain the corresponding
quark and electron chemical potentials. In Fig. 1, quark and electron chemical
potentials are shown as functions of the baryon density at fixed $B_{m}$. The
blue dashed and green dotted curves correspond to $\mu_{u}$ and $\mu$ with the
same magnetic field strength $B_{m}=10^{18}$ G, while the red and black solid
curves represent the electron chemical potentials at two different magnetic
field strengths $B_{m}=10^{17}$ G and $B_{m}=10^{18}$ G, respectively. As can
be seen from the figure, both $\mu_{u}$ and $\mu$ monotonically increase with
baryon density for fixed magnetic field strength. While in both cases the
electron chemical potential $\mu_{e}$, first increases then decrease after
reaching a maximum with increasing baryon density. And the peaks for the
electron chemical potential are approximately located at $n_{b}\approx
0.20~{}\text{fm}^{-3}$. The only difference of these two curves is the obvious
oscillation shown by the solid black line with $B_{m}=10^{18}$ G, an order
larger than $B_{m}=10^{17}$ indicated with the solid red line. As a minor
comment, we notice that $\mu_{e}$ experiences fluctuations for large values of
$B_{m}$. These are due to the fact that electron density is fixed by the
condition of electrical neutrality: at zero temperature and in the lowest
Landau level approximation, $n_{e}\propto eB_{m}\mu_{e}$; thus, $\mu_{e}$ is
sensitive to the behavior of the density of the quarks and in particular to
their oscillations. However, the oscillations of $\mu_{e}$ as well as those of
the chemical potentials of the quarks are very tiny, see the scale on the
right vertical axis in Fig. 1: they are of the order of a few MeV, therefore
they are easy to visualize for the $\mu_{e}$ which is of the order of 20 MeV,
but are invisible for the quarks since their chemical potentials are in the
range $200\sim 500$ MeV.

Except for the chemical potentials, the presence of a magnetic field also
modifies the distribution of Landau levels. In Fig. 2, the maximum Landau
levels for each component of SQM, i.e. up, down, strange quarks and electrons,
are plotted as functions of the baryon density for two different magnetic
field strengths. The red and black solid curves represent the cases with
magnetic field strength $B_{m}=10^{17}$ G and $B_{m}=10^{18}$ G, respectively.
We observe that the curves, representing the maximum Landau levels of quarks,
grow almost linearly with increasing density at fixed magnetic field strength.
Owing to the strong suppression of $\nu_{i,\text{max}}$ in strong magnetic
fields, the slope of the black curves ($B_{m}=10^{18}$ G) is much smaller than
that of the red curves ($B_{m}=10^{17}$ G). Moreover, at very low densities
with fixed magnetic field strength, the inequalities
$\nu_{d,\text{max}}>\nu_{u,\text{max}}>\nu_{s,\text{max}}$ hold, while at
relatively high density, we have
$\nu_{d,\text{max}}>\nu_{s,\text{max}}>\nu_{u,\text{max}}$. This phenomenon
can be understood because down and strange quarks share the same chemical
potential $\mu$, but the latter always has a larger effective mass due to the
non-vanishing current mass $m_{s0}$, which automatically leads to
$\nu_{d,\text{max}}>\nu_{s,\text{max}}$. On the other hand, the condition
$\mu_{u}<\mu\equiv\mu_{s}=\mu_{d}$ should be fulfilled for any density because
of the consideration of $\beta$-equilibrium with the weak reactions
$d,s\leftrightarrow u+e+\bar{v}_{e}$ and $s+u\leftrightarrow u+d$. As a
consequence, $\nu_{d,\text{max}}>\nu_{u,\text{max}}>\nu_{s,\text{max}}$ holds
at low densities owing to the large value of effective strange quark mass but
small strange quark chemical potential in this region. However, as we increase
the density, the numerator in Eq. (7) for the strange quark increases much
faster than the one for the up quark, and at the same time
$\nu_{u,\text{max}}$ is always reduced by a factor of two compared to
expression of $\nu_{s,\text{max}}$ due to the ratio $|q_{u}/q_{s}|=2$. It can
be checked that as the density increases, the difference between these two
maximal Landau levels grows and finally leads to
$\nu_{u,\text{max}}<\nu_{s,\text{max}}$ at high densities.

For the two selected magnetic field strengths, i.e., $B_{m}=10^{17}$ G and
$B_{m}=10^{18}$ G, the maximum Landau levels for electrons keep zero since the
corresponding electron chemical potentials are small, which are only a few
tens MeV, compared with the magnetic field strengths. In this case, only the
lowest Landau level contributes to the thermodynamic quantities of magnetized
SQM, which is also explicitly confirmed in Fig. 3. This particular shape for
the density dependence of the electron chemical potential is responsible for
the behavior of the relative spin polarization $\Delta_{e}$ as shown in the
two panels of Fig. 5. From Eq. (11), it is expected that the number density of
electrons is completely determined by the behavior of the electron chemical
potential $\mu_{e}$ for a fixed magnetic field strength, namely, $n_{e}$ also
first increases and then decreases with increasing baryon density.

Figure 3: (Color online) The maximum Landau levels for various charged
particles as a function of magnetic field strength for $n_{b}=2n_{0}$ (left
panel) and $n_{b}=4n_{0}$ (right panel), where $n_{0}=0.16~{}\text{fm}^{-3}$
is the saturation nuclear density.

In Fig. 3, we plot the maximum Landau levels for each component of
$\beta$-equilibrium magnetized SQM as functions of the magnetic field strength
at $n_{b}=2n_{0}$ (left panel) and $n_{b}=4n_{0}$ (right panel), where
$n_{0}=0.16~{}\text{fm}^{-3}$ is the saturation density of normal nuclear
matter. For the sake of convenience we use the logarithm to label both the
vertical and horizonal axes. The electron maximum Landau level
$\nu_{e,\text{max}}$, represented by the black dotted curve, decreases from a
value about ten to one at around $1.8\times 10^{16}~{}\text{G}$ and $1.5\times
10^{16}~{}\text{G}$ in the left and right panels respectively. While the
maximum Landau levels for down, strange, and up quarks, which are denoted by
the solid blue, solid red, and dashed green curves from right to left
respectively, exhibit similar behavior: the curves decrease monotonously with
increasing magnetic field strength and show obvious ladder-like shape at
extremely large magnetic field strengths. Furthermore, as can be seen that the
maximum Landau levels decrease almost linearly with the magnetic field
strength, which agrees well with the Nambu$-$Jona-Lasinio finding of Ref. Wen
and Liang (2016).

Figure 4: (Color online) Quark fractions, $n_{u}/(3n_{b})$, $n_{d}/(3n_{b})$,
$n_{s}/(3n_{b})$ and $10^{3}$ times electron fraction $1000n_{e}/(3n_{b})$, as
functions of the baryon density for $B_{m}=10^{17}$ G (left panel) and
$B_{m}=10^{18}$ G (right panel) respectively.

In Fig. 4, we plot the quark fractions, i.e. $n_{u}/(3n_{b})$,
$n_{d}/(3n_{b})$, $n_{s}/(3n_{b})$, and the $10^{3}$ times electron fraction
$1000n_{e}/(3n_{b})$, as functions of the baryon density at fixed magnetic
field strength $B_{m}=10^{17}$ G (left panel) and $B_{m}=10^{18}$ G (right
panel), respectively. One can see that the fraction of down and strange quarks
decrease and increase with increasing baryon density respectively in both
panels, while the fraction of up quarks almost keep constant in the considered
range of the baryon density. The fractions of different quark flavors approach
each other when the density is large enough. In addition, the fraction of
electrons is very small and it decreases with increasing density. Also, the
oscillation of the electron fraction become more obvious for larger magnetic
field strengths.

###  III.2 Spin polarization

Figure 5: (Color online) Variation of the relative spin polarization for up,
down, strange quarks, and electron with respect to the magnetic field strength
at $n_{b}=2n_{0}$ (left panel) and $n_{b}=4n_{0}$ (right panel).

From Eq. (11), one can easily deduce that in the presence of an external
magnetic field the number density of each constituent of magnetized SQM has
two contributions, corresponding to the particles with spin parallel and
antiparallel to the magnetic field orientation respectively. To study the spin
polarization of the magnetized SQM, we introduce the relative spin
polarization for $i$-type particle as Perez-Garcia, Providencia, and Rabhi
(2011); Avancini, Menezes, and Providencia (2011); Rabhi  _et al._ (2015)

$\displaystyle\Delta_{i}=\frac{n_{i}^{\uparrow}-n_{i}^{\downarrow}}{n_{i}^{\uparrow}+n_{i}^{\downarrow}},$
(19)

where $n_{i}^{\uparrow}$ and $n_{i}^{\downarrow}$ denote the number density of
spin up and down $i$-type particles. When $B_{m}=0$, we obviously have
$n_{i}^{\uparrow}=n_{i}^{\downarrow}$ and hence $\Delta_{i}=0$. For nonzero
external magnetic field, however, there is an upper value of the magnetic
field $B_{m}^{c,i}$ for which complete saturation of each constituent of the
SQM occurs according to the expression
$n_{i}=n_{i}^{\uparrow}+n_{i}^{\downarrow}$ and Eq. (11). When the magnetic
field reaches or excess the critical value $B_{m}^{c,i}$, from Eq. (5) the
condition for each constituent particle $\Delta_{i}=1$ or $\Delta_{i}=-1$ is
satisfied. The critical field value $B_{m}^{c,i}$ saturates the system and
aligns all the $i$-type particles parallel or antiparallel to the magnetic
field depending on the electric charge of the particles. Note that as deduced
from the MIT bag model Felipe  _et al._ (2008); Felipe and Martinez (2009)
without taking into account the strong interactions between quarks, the
inclusion of anomalous magnetic moment will further affects the properties of
magnetized SQM and hence alerts the values of $B_{m}^{c,i}$, but which is
beyond the scope of this work.

In Fig. 5, we plot the relative spin polarization of each constituent of
magnetized SQM as a function of the magnetic field strength, for two different
baryon densities $n_{b}=2n_{0}$ (left panel) and $n_{b}=4n_{0}$ (right panel)
respectively. The curves from top to bottom correspond to $\Delta_{u}$ (green
dashed curve), $\Delta_{d}$ (blue solid curve), $\Delta_{s}$ (red dot-dashed
curve), and $\Delta_{e}$ (black dotted curve) in both panels. We observe that
the curves in both panels show similar behaviors for each type particle with
increasing magnetic field strength. More specifically, the relative spin
polarization of electrons $\Delta_{e}$ begins to depart from zero at around
$10^{15}$ G and the electrons become completely align antiparallel to the
magnetic field at around $3.6\times 10^{16}$ G and $3.0\times 10^{16}$ G in
the left and right panels, respectively. While the relative spin polarization
of quarks begins to departure from each other at around $3.0\times 10^{17}$ G
and this effect becomes notable after $10^{18}$ G. As shown in the left panel
of Fig. 5, $\Delta_{u}$, $\Delta_{s}$, and $\Delta_{d}$ successively become
completely saturated at the magnetic field strengths $1.3\times 10^{19}$ G,
$1.7\times 10^{19}$ G, and $2.5\times 10^{19}$ G. And a comparison of these
two panels shows that for a larger baryon density the critical magnetic field
$B_{m}^{c,i}$ for quarks becomes larger, whereas the one for electrons does
exactly the opposite. As stated in Sec. III.1, this behavior can be explained
by the density dependence of the quark and electron chemical potentials shown
in Fig. 1. Namely, due to the monotonic behavior of quark chemical potentials
as functions of the density, larger quark chemical potentials lead to larger
maximum Landau levels and in turn to smaller absolute values of $\Delta_{i}$
for quarks. However, for the two chosen baryon densities $n_{b}=2n_{0}$ and
$n_{b}=4n_{0}$, which are larger than typical value of the location of the
peak for electron chemical potential $n_{b}\approx 0.20~{}\text{fm}^{-3}$
shown in Fig. 1, the electron chemical potential decreases with increasing
$n_{b}$, which means that a larger baryon density $n_{b}$ leads to a smaller
maximum Landau level $\nu_{e,\text{max}}$ and hence to a smaller absolute
value of $\Delta_{e}$.

Our study reveals that there is a narrow region, close to the critical field
value that $\Delta_{u}=-1$ is reached, in which the $\beta$-equilibrium and
charge neutrality conditions can not be fulfilled simultaneously. In other
words, a negative solution for the electron chemical potential is found in
this region Isayev and Yang (2013). When the magnetic field becomes larger,
the charge neutrality condition is fulfilled again. Besides, one can also read
from the figure that depending on the electric charge, the up quarks tend to
polarize aligning spins with the magnetic field orientation, whereas the
particles with negative electric charges like electrons, down and strange
quarks do the opposite, which is consistent with the finding in Refs. Perez-
Garcia, Providencia, and Rabhi (2011); Avancini, Menezes, and Providencia
(2011).

Figure 6: (Color online) Relative spin polarization of up, down, and strange
quarks in $\beta$-equilibrium as a function of the baryon density for
$B_{m}=10^{17}$ G (red dotted curves) and $B_{m}=5\times 10^{18}$ G (black
solid curves) respectively.

In Fig. 6, we display the variation of the relative spin polarization with
density for up, down, and strange quarks under the $\beta$-equilibrium
condition at two representative magnetic field strengths $B_{m}=10^{17}$ G and
$B_{m}=5\times 10^{18}$ G. For the magnetic field strength $B_{m}=10^{17}$ G,
denoted by the red dotted curves, the number densities with spin up and down
almost coincide with each other in the whole considered density range. While
for the magnetic field strength $B_{m}=5\times 10^{18}$ G, denoted by the
black solid curves, all the curves largely deviate from zero at low densities
and gradually goes to zero with increasing baryon density. As we have shown in
Figs. 2 and 3 that the maximum Landau levels of quarks increase with the
increase of the baryon density but decrease with increasing magnetic field
strength. As a consequence, the characteristic behavior of the occupation of
the Landau levels responsible for that although the magnetic field effects
enhance the spin polarization, the finite baryon density effect tends to
reduce the difference of spin polarization for magnetized SQM.

###  III.3 Sound velocity

Figure 7: (Color online) Sound velocity of magnetized cold quark matter as a
function of the baryon density for different values of magnetic fields. The
horizontal line represents the conformal limit $V_{s}^{2}=1/3$.

The strength of interaction and the stiffness of the equation of state can be
measured by the sound velocity. In Fig. 7, we present the sound velocity
squared $V_{s}^{2}$ as a function of the baryon density, which is determined
by

$\displaystyle V_{s}^{2}=\frac{\mathrm{d}P}{\mathrm{d}E}.$ (20)

The red and black solid curves correspond to $B_{m}=10^{17}$ G and
$B_{m}=10^{18}$ G respectively, while the black dashed line represents the
conformal limit. It is found that in both cases, the sound velocity increase
from small values with increasing baryon density and asymptotically approaches
the conformal limit $V_{s}^{2}=1/3$ at high densities. The only difference is
that the curve with a stronger magnetic field shows a more obvious
oscillation. Except for the oscillation, the density behavior of sound
velocity in the present work is qualitatively the same as the zero magnetic
field case Wen  _et al._ (2009). Since the two curves approaches the conformal
limit from below, the sound velocity of magnetized SQM also fulfills the
causality limit $V_{s}^{2}<1$. It would be worth mentioning that to explain
the recent available cosmological data like large mass of the pulsars with a
mass of about two solar masses and tidal deformability observational data,
there have been many discussions suggesting that the sound velocity of dense
matter should larger than the conformal limit $V_{s}^{2}=1/3$, see e.g. Refs.
Alsing, Silva, and Berti (2018); Tews, Margueron, and Reddy (2018); Reed and
Horowitz (2020). Such statements have been confirmed both by the model
calculations Xia  _et al._ (2021) and statistical analysis Li  _et al._
(2021). However, we should emphasize that such kind of calculations are mostly
done by assuming that the compact stars are neutron stars or hybrid stars
undergoing a phase transition Hippert, Fraga, and Noronha (2021).

##  IV Conclusions

We have explored the behaviors of various thermodynamic quantities of the
magnetized SQM, with the consideration of $\beta$-equilibrium and charge
neutrality conditions, to the baryon density as well as to the magnetic field
strength at nonzero chemical potentials within the framework of the
quasiparticle model. The quark chemical potentials are found to increase with
the increment of baryon density at a certain magnetic field strength, which
has the same behavior as the zero magnetic field case.

As the magnetic field increases, the relative spin polarization of electrons
and quarks begin to depart from zero at $\mathcal{O}(10^{15}~{}\text{G})$ and
$\mathcal{O}(10^{17}~{}\text{G})$, and the corresponding absolute values
become one at $\mathcal{O}(10^{16}~{}\text{G})$ and
$\mathcal{O}(10^{19}~{}\text{G})$, respectively. In particular, for higher
values of the baryon density the critical magnetic field strengths for
$|\Delta_{i=u,d,s}|=1$ moves to larger values of $B_{m}$, while the one for
electrons does the opposite or the same depending on the location of the
corresponding electron chemical potential as shown in Fig. 1. Namely, a
smaller value of $\mu_{e}$ or $n_{e}$ leads to a larger $|\Delta_{e}|$ and in
turn to a smaller critical magnetic field when $\Delta_{e}=-1$ is satisfied.
Increasing the baryon density or the magnetic field affects the net
polarization of quarks differently: this is easy to understand in terms of the
occupation of the Landau levels, see Eq. (7). In fact, from that equation we
see that increasing the chemical potential while keeping $B_{m}$ fixed results
in a higher number of occupied Landau levels, thus reducing the polarization.
On the other hand, increasing $B_{m}$ while keeping the chemical potential
fixed, results in the lowering of the number of occupied Landau levels hence
in the increase of the net polarization of strange quark matter. Thus, one can
conclude that as a consequence of the occupation of Landau levels that in
contrast to the finite baryon density effect which reduces the absolute value
of relative spin polarization, the magnetic field effect leads to an
enhancement of spin polarization of the magnetized SQM.

We have analyzed, for the first time within the quasiparticle model, the
polarization of strange quark matter in the background of a strong magnetic
field; moreover, we have also computed the squared speed of sound. Compared to
the zero-magnetic field case, the sound velocity in an external magnetic field
shows an obvious oscillation with increasing density, especially for the curve
with a stronger magnetic field. The sound velocity, in addition to the visible
oscillation phenomena, grows with increasing density and approaches the
conformal limit $V_{s}^{2}=1/3$ at high densities from below.

## Acknowledgments

The authors thank the support from the National Natural Science Foundation of
China (Grant Nos. 11875181, 11875052, 11947098, 12005005, and 61973109), the
Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ40188),
the Scientific Research Fund of Hunan Provincial Education Department of China
(Grant No. 19C0772), and the Scientific Research Fund of Hunan University of
Science and Technology (Grant No. E52059). M. R. is supported by the National
Natural Science Foundation of China (Grants No. 11805087 and No. 11875153) and
by the Fundamental Research Funds for the Central Universities (Grant No.
862946).

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