# Shear-induced spin polarization in heavy-ion collisions

Baochi<EMAIL_ADDRESS>Department of Physics & State Key Laboratory
of Nuclear Physics and Technology,
Peking University, Beijing 100871, China  Collaborative Innovation Center of
Quantum Matter, Beijing 100871, China  Shuai Y. F<EMAIL_ADDRESS>Quark Matter Research Center, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  Longgang<EMAIL_ADDRESS>Key
Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Huichao
<EMAIL_ADDRESS>Department of Physics & State Key Laboratory of
Nuclear Physics and Technology,
Peking University, Beijing 100871, China  Collaborative Innovation Center of
Quantum Matter, Beijing 100871, China  Center for High Energy Physics, Peking
University, Beijing 100871, China  Yi<EMAIL_ADDRESS>Quark Matter
Research Center, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

We study the spin polarization generated by the hydrodynamic gradients. In
addition to the widely studied thermal vorticity effects, we identify an
undiscovered contribution from the fluid shear. This shear-induced
polarization (SIP) can be viewed as the fluid analog of strain-induced
polarization observed in elastic and nematic materials. We obtain the explicit
expression for SIP using the quantum kinetic equation and linear response
theory. Based on a realistic hydrodynamic model, we compute the differential
spin polarization along both the beam direction $\hat{z}$ and the out-plane
direction $\hat{y}$ in non-central heavy-ion collisions at $\sqrt{s_{NN}}=200$
GeV, including both SIP and thermal vorticity effects. We find that SIP
contribution always shows the same azimuthal angle dependence as experimental
data and competes with thermal vorticity effects. In the scenario that
$\Lambda$ inherits and memorizes the spin polarization of strange quark, SIP
wins the competition, and the resulting azimuthal angle dependent spin
polarization $P_{y}$ and $P_{z}$ agrees qualitatively with the experimental
data.

Introduction.—The transport phenomena involving spin are instrumental in
investigating quantum effects in many-body systems. For example, the
generation of spin current can be employed to probe intriguing properties of
quantum materials Han  _et al._ (2019). In relativistic heavy-ion collisions,
measuring spin polarization of hyperons has been proposed to explore the spin
dynamics of quarks in the produced quark-gluon plasma (QGP) in heavy-ion
collisions Liang and Wang (2005). The observed $\Lambda$ spin polarization at
RHIC and LHC experiments Adamczyk  _et al._ (2017); Adam  _et al._ (2018,
2019); Niida (2019) opens a new avenue to study the hot and dense nuclear
matterBecattini  _et al._ (2013); Fang  _et al._ (2016); Karpenko and
Becattini (2017); Pang  _et al._ (2016); Becattini and Karpenko (2018); Li
_et al._ (2017); Becattini  _et al._ (2019); Liu  _et al._ (2020); Becattini
and Lisa (2020); Huang  _et al._ (2020); Florkowski  _et al._ (2018); Hattori
_et al._ (2019a); Fukushima and Pu (2020); Li  _et al._ (2020); Bhadury  _et
al._ (2021); Li and Yee (2019); Wang  _et al._ (2020); Yang  _et al._ (2020);
Weickgenannt  _et al._ (2020, 2021); Li and Liu (2020); Sheng  _et al._
(2020); Wang and Zhuang (2021); Singh  _et al._ (2020); Wu  _et al._ (2019);
Fu  _et al._ (2020); Florkowski  _et al._ (2019a, b); Sun and Ko (2017, 2019);
Xia  _et al._ (2018); Zhang  _et al._ (2019); Speranza and Weickgenannt
(2020); Csernai  _et al._ (2019); Becattini (2020); Liu and Yin (2020, 2021).

During the hydrodynamic evolution of the fireball created in heavy-ion
collisions, the sizable gradients of hydrodynamic fields, such as temperature
and flow gradient, could give rise to spin polarization. One widely-studied
effect is the spin polarization induced by thermal vorticity Becattini  _et
al._ (2013); Fang  _et al._ (2016); Pang  _et al._ (2016), which is a
combination of temperature gradient and fluid vorticity. These theories based
on the thermal vorticity effects successfully describe the global $\Lambda$
polarization measured in experiments Adamczyk  _et al._ (2017); Adam  _et al._
(2018). In contrast, the predicted local (differential) $\Lambda$ spin
polarization Becattini and Karpenko (2018); Xia  _et al._ (2018); Fu  _et al._
(2020) differs qualitatively from experimental observations Niida (2019); Adam
_et al._ (2019), see attempts to address such “spin sign puzzle” in Refs. Liu
_et al._ (2020); Wu  _et al._ (2019); Florkowski  _et al._ (2019b).

Nevertheless, vorticity and temperature gradient are not the only examples of
hydrodynamic gradients. In this letter, we identify the missing contribution
to the spin polarization, namely, the effect of the shear stress tensor
$\sigma^{\mu\nu}$, or fluid shear. This shear-induced polarization (SIP) is
not only allowed by symmetry but can also be derived explicitly based on
quantum kinetic equation and the linear response theory, as we shall
demonstrate later. SIP can be viewed as the fluid analog of strain-induced
polarization observed in elastic and nematic materials Crooker and Smith
(2005); Mal’Shukov  _et al._ (2005).

Employing the data-calibrated hydrodynamic calculation Fu  _et al._ (2020), we
investigate spin polarization, including both SIP and thermal vorticity
effects. We find in the scenario that $\Lambda$ inherits and memorizes the
spin polarization of strange quark, SIP wins over the effects of thermal
vorticity. As such, the total spin polarization shows an azimuthal angle
dependence qualitatively agrees with the experimental data.

Theory.—Considering a system in the presence of slow varying flow velocity
$u^{\mu}$ and temperature $T$, we look for gradient expansion of axial Wigner
function ${\cal A}^{\mu}$ which describes the phase space density of spin
polarization of fermions.

Figure 1:  Spin polarization as a function of azimuthal angle $\phi_{p}$ along
$z$ and $y$ directions, induced by hydrodynamic gradients for $\Lambda$
hyperon and strange quarks at the freezeout surface: colored curves show
effects from vorticity, temperature gradient and shear stress tensor (i.e. the
shear-induced polarization or SIP) , corresponding to the first, second and
third terms in Eq. (Shear-induced spin polarization in heavy-ion collisions)
respectively. The effects of thermal vorticity are given by the sum of
vorticity and temperature gradient effects and have been studied by many
others. SIP is the new effect studied in this letter and competes with the
thermal vorticity effects.

We will first study the simplest case that fermions are massless using the
expression from chiral kinetic theory Son and Yamamoto (2012); Chen  _et al._
(2014, 2015):

$\displaystyle\,{\cal
A}^{\mu}=\sum_{\lambda}\,\left(\lambda\,p^{\mu}\,f_{\lambda}+\frac{1}{2}\frac{\epsilon^{\mu\nu\alpha\rho}p_{\nu}u_{\alpha}\partial_{\rho}f_{\lambda}}{p\cdot
u}\right)\,,$ (1)

where $\lambda=+/-$ accounts for right/left-handed chiral fermions
respectively, $p_{\mu}$ is the momentum and $f_{\lambda}$ is the distribution
function of particles. The second term in Eq. (1) is commonly referred as
magnetic current MC term Hattori  _et al._ (2019b) which gives rise to various
interesting transport phenomena such as off-equilibrium chiral magnetic effect
Kharzeev  _et al._ (2017), spin Hall effect Son and Yamamoto (2013); Hattori
_et al._ (2019b); Liu and Yin (2020), and has been recently implemented into
transport model for spin polarization Liu  _et al._ (2020).

Now, we replace $f_{\lambda}$ in Eq. (1) with local equilibrium distribution
function $n(\beta(\varepsilon_{0}-\Delta\varepsilon_{\lambda}))$ where the
energy shift due to spin-vorticity coupling is given by
$\Delta\varepsilon_{\lambda}=-(1/2)\lambda\omega\cdot p/\varepsilon_{0}$ with
$\varepsilon_{0}=p\cdot u$ and
$\omega^{\mu}=(1/2)\epsilon^{\mu\nu\alpha\lambda}u_{\nu}\partial^{\perp}_{\alpha}u_{\lambda}$
is the vorticity. Here, $n(x)=1/(e^{x}+1)$ denotes Fermi-Dirac function. Then,
we can expand Eq. (1) to first order in gradient as

$\displaystyle{\cal A}^{\mu}=\beta n_{0}(1-n_{0})\Big{[}$
$\displaystyle-\frac{1}{\varepsilon_{0}}p^{\mu}p^{\nu}\omega_{\nu}+\epsilon^{\mu\nu\alpha\rho}u_{\nu}\,p_{\alpha}\beta^{-1}\partial_{\rho}\beta$
$\displaystyle-$
$\displaystyle\frac{1}{\varepsilon_{0}}\epsilon^{\mu\nu\alpha\rho}u_{\nu}\,p_{\rho}p^{\lambda}\partial^{\perp}_{\alpha}u_{\lambda}\Big{]}\,.$
(2)

Here, we denote a generic vector $V^{\mu}$ projected along the transverse
direction with respect to $u^{\mu}$ as
$V^{\mu}_{\perp}=\Delta^{\mu\nu}V_{\nu}$ where
$\Delta^{\mu\nu}=\eta^{\mu\nu}-u^{\mu}u^{\nu}$ and
$\eta^{\mu\nu}=(1,-1,-1,-1)$ is the metric and $\epsilon^{0123}=1$. We have
also introduced the notation $n_{0}\equiv n(\beta\varepsilon_{0})$. To
proceed, we use
$\partial^{\perp}_{\alpha}u_{\lambda}=\epsilon_{\alpha\lambda\gamma\zeta}u^{\gamma}\omega^{\zeta}+\partial^{\perp}_{(\alpha}u_{\lambda)}$
to evaluate the last term in Eq. (Shear-induced spin polarization in heavy-ion
collisions) where
$\partial^{\perp}_{(\alpha}u_{\lambda)}\equiv(\partial^{\perp}_{\alpha}u_{\lambda}+\partial^{\perp}_{\lambda}u_{\alpha})/2$.
Noting
$(\,-p_{\perp}\cdot\omega)\,u^{\mu}+\varepsilon_{0}\,\omega^{\mu}=(1/2)\epsilon^{\mu\nu\alpha\lambda}p_{\nu}\partial^{\perp}_{\alpha}u_{\lambda}$
and $p^{2}=0$ for massless fermions 666We have used $p^{2}=0$ to make the
connection between the results from massless case and that of massive theory
more transparent., we arrive at the desired expression

$\displaystyle{\cal A}^{\mu}=$ $\displaystyle\frac{1}{2}\beta
n_{0}(1-n_{0})\,\Big{\\{}\epsilon^{\mu\nu\alpha\lambda}p_{\nu}\partial^{\perp}_{\alpha}u_{\lambda}$
$\displaystyle+2\epsilon^{\mu\nu\alpha\lambda}u_{\nu}p_{\alpha}\,[\beta^{-1}(\partial_{\lambda}\beta)]\Big{\\}}+{\cal
A}^{\mu}_{\text{SIP}}\,$ (3)

where the shear-induced spin polarization (SIP) as we advertised earlier is
given by

$\displaystyle{\cal A}^{\mu}_{\text{SIP}}$ $\displaystyle=-\beta
n_{0}(1-n_{0})\frac{1}{\varepsilon_{0}}\epsilon^{\mu\nu\alpha\rho}u_{\nu}\,p_{\rho}p^{\lambda}\partial^{\perp}_{(\alpha}u_{\lambda)}$
$\displaystyle=-\beta
n_{0}(1-n_{0})\,\frac{p^{2}_{\perp}}{\varepsilon_{0}}\epsilon^{\mu\nu\alpha\rho}u_{\nu}Q_{\alpha}^{\,\,\lambda}\sigma_{\rho\lambda}\,.$
(4)

From the first line to the second line in Eq. (Shear-induced spin polarization
in heavy-ion collisions), we have used
$p^{\mu}=\varepsilon_{0}u^{\mu}+p^{\mu}_{\perp}$. The generalized quadrupole
tensor and shear stress tensor are given by
$Q^{\mu\nu}\equiv-p^{\mu}_{\perp}p^{\nu}_{\perp}/p^{2}_{\perp}+\Delta^{\mu\nu}/3$
and
$\sigma^{\mu\nu}=\partial^{(\mu}_{\perp}u^{\nu)}_{{{\color[rgb]{1,1,1}\perp}}}-\Delta^{\mu\nu}\partial\cdot
u/3$ respectively. Note ${\cal A}^{\mu}_{\text{SIP}}$ solely arises from the
magnetic current term.

To extend our analysis to fermions with an arbitrary mass, we consider the
linear response theory (detailed in Ref Liu and Yin (2021)) and find that the
Eq. (Shear-induced spin polarization in heavy-ion collisions) applies equally
to massless and massive fermions. This means that the axial Wigner function
${\cal A}^{\mu}$ only inexplicitly depends on the fermion mass through the
mass dependence of $p^{\mu}$. The key in the linear response analysis is to
match the gradient expansion of ${\cal A}^{\mu}$ to small frequency and
wavevector behavior of the retarded correlator
$G^{\alpha;\mu\nu}_{R}=\langle\bar{\psi}(t,\bm{x}-\frac{\bm{y}}{2})\,\gamma^{\alpha}\gamma^{5}\psi(t,\bm{x}+\frac{\bm{y}}{2})\,\hat{T}^{\mu\nu}(0,{\bm{0}})\rangle\theta(t)$
where $\hat{T}^{\mu\nu}$ denotes the stress-energy tensor and $\psi$ is the
fermionic field. Using one loop calculation of $G^{\alpha;\mu\nu}_{R}$, we
confirm Eq. (Shear-induced spin polarization in heavy-ion collisions) for
fermions with arbitrary mass.

For the convenience of the subsequent discussion, we combine the first two
terms in “{ }” in Eq. (Shear-induced spin polarization in heavy-ion
collisions) using the hydrodynamic equation
$(u\cdot\partial)u_{\alpha}=-\beta^{-1}\partial^{\perp}_{\alpha}\beta+{\cal
O}(\partial^{2})$ to obtain

$\displaystyle{\cal A}^{\mu}$
$\displaystyle=\frac{1}{2}n_{0}(1-n_{0})\,\epsilon^{\mu\nu\alpha\lambda}p_{\nu}\partial_{\alpha}(\beta
u_{\lambda})+A^{\mu}_{\rm SIP}\,$ (5)

where the first term reproduces the spin polarization induced by thermal
vorticity, as was studied by many authors Fang  _et al._ (2016); Pang  _et
al._ (2016); Becattini  _et al._ (2013). What is our new finding is the second
term which describes the effects of shear stress tensor on spin polarization.

Figure 2:  Spin polarization of $\Lambda$ hyperon (left) and strange quark
(middle) along $z$ (upper) and $y$(lower) directions induced by the combined
effects of shear stress tensor and thermal vorticity (solid curves) and by
thermal vorticity effects only (dashed curves) on the freezeout surface.
Right: the re-plotted experimental data in Ref Niida (2019); Adam  _et al._
(2019). The $P_{z}$ is converted using
$P_{z}=\langle\cos\theta^{*}_{p}\rangle/[\alpha_{H}\langle(\cos\theta^{*}_{p})^{2}\rangle]$
assuming zero error in denominator. The results for strange spin polarization
illustrate the anticipated qualitative behavior in the ”strange
memory”scenario, i.e., the memory of strange quark polarization is preserved
in the measured $\Lambda$ polarization.

Two scenarios.—To quantitatively predict the $\Lambda$ spin polarization due
to hydrodynamic gradient effects, we need to hadronize the polarized strange
quarks into Lambda hyperons followed by hadronic evolution towards the kinetic
freezeout. However, the development of hadronization and transport models
which consistently include the spin degrees of freedom is highly non-trivial,
see Refs. Sheng  _et al._ (2020); Wang and Zhuang (2021) for recent
progresses. In order not to introduce complicated model assumptions, we shall
consider two widely assumed limiting scenarios Karpenko and Becattini (2017);
Pang  _et al._ (2016); Becattini and Karpenko (2018); Fu  _et al._ (2020);
Liang and Wang (2005); Sun and Ko (2017, 2019) and focus not on the magnitude
but the qualitative features of the resulting azimuthal angle dependence of
spin polarization.

In the first scenario, namely the “Lambda equilibrium”, we shall assume the
spin relaxation rate is large enough so that $\Lambda$ hyperons immediately
response to the presence of hydrodynamic gradients once $\Lambda$ are formed
through hadronization. In the second scenario, we consider the opposite limit
that $\Lambda$ “inherits” the spin polarization from its constituent strange
quark Jennings (1990); Cohen and Weber (1991), and the resulting spin
polarization is frozen ever since the hadronization. This scenario will be
referred to as the “strange memory”. In reality, $\Lambda$ spin polarization
should evolve from the “strange memory” scenario towards that in the ”Lambda
equilibrium” scenario. Therefore comparing results from those two benchmark
scenarios provides us qualitative guidance on what we anticipate to observe in
heavy-ion collisions.

Guided by Refs Becattini  _et al._ (2013); Fang  _et al._ (2016), we shall use
the following freezeout prescription to connect axial Wigner function given by
Eq. (Shear-induced spin polarization in heavy-ion collisions) to spin
polarization vector $P^{\mu}$ on the freezeout hyper-surface $\Sigma_{\mu}$ :

$\displaystyle P^{\mu}(\bm{p})=\frac{\displaystyle\int
d\Sigma^{\alpha}p_{\alpha}\,{\cal A}^{\mu}(x,\bm{p};m)}{\displaystyle 2m\int
d\Sigma^{\alpha}p_{\alpha}n(\beta\varepsilon_{0})}\,.$ (6)

Here the factor of $2$ in the denominator counts two states of the spin-$1/2$
fermions. In the “Lambda equilibrium” (“strange memory”) scenario, we shall
compute $\Lambda$ (strange quark) spin polarization using Eq. (6). We use
$m=1.116$ GeV for Lambda mass and the benchmark value for the strange quark
mass $m=0.3$ GeV, respectively; the latter is between strange constituent and
current mass. In principle, we should compute Eq. (6) at the kinetic freezeout
for “Lambda equilibrium” scenario, but we have checked that the results of
doing so are qualitatively similar to those calculated at chemical freezeout.
To simplify the comparison, we shall show spin polarization vector computed at
chemical freezeout in both scenarios.

Results– In this letter, we implement 3+1-d hydrodynamics MUSIC Schenke  _et
al._ (2010, 2011, 2012) with AMPT initial conditions Lin  _et al._ (2005);
Pang:2012he; Xu  _et al._ (2016) to generate the freeze-out surface and
associated temperature and flow velocity profiles for the spin polarization
calculation described by Eq.(3). Unless noted otherwise, we use the same
inputs and parameter sets as used in previous paper Fu  _et al._ (2020) where
the “Lambda equilibrium” scenario without SIP has been studied. Such
hydrodynamic calculations have been well calibrated to fit the
$dN_{\text{ch}}/dy$, $p_{T}$ spectra and $v_{2}(p_{T})$ of pions and protons
in Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV. 777 We have also performed an
independent calculation using CLVisc Pang  _et al._ (2016) hydrodynamics
framework and produce all the key features shown in the main text. . In what
follows, we will focus on the spin polarization vector along the beam
direction, $P_{z}(\phi_{p})$, and along the out-plane direction,
$P_{y}(\phi_{p})$, as a function of the azimuthal angle $\phi_{p}$.

In Fig. 1 and Fig. 2, we plot the differential spin polarization vector along
the beam direction, $P_{z}(\phi_{p})$, and along the out-plane direction,
$P_{y}(\phi_{p})$, in the particle rest frame. They include the separate
and/or combined effects from SIP, fluid vorticity and temperature gradient.
Although the thermal vorticity contribution has been studied extensively, this
is the first time that the signature of SIP has been investigated in heavy-ion
collisions. All curves contributing to $P_{z}$ and $(-P_{y})$ can be
parametrized approximately as “$b_{z}\sin(2\phi_{p})$” and
“$a+b_{y}\cos(2\phi_{p})$” respectively. For the transparency of comparing
with the qualitative features of the experimental results shown in Fig. 2 , we
shall refer a contribution as the ”same” sign (“opposite” sign) when
$b_{z},b_{y}>0$ ($b_{z},b_{y}<0$).

In Fig. 1, we plot spin polarization induced by shear stress tensor (SIP),
vorticity (VoIP) and temperature gradient (TIP) (c.f. Eq. (Shear-induced spin
polarization in heavy-ion collisions)) in both “Lambda equilibrium” and
“strange memory” scenarios. We observe that the contribution from SIP and TIP
to global polarization is insignificant. However, the azimuthal angle
dependence of $P_{z}$ and $P_{y}$ arises mostly from SIP and TIP. As a marked
qualitative feature, SIP always leads to the “same” sign contribution to spin
polarization in both $z$ and $y$ directions. On the other hand, the effects of
thermal vorticity on the azimuthal angle dependence of spin polarization are
dominated by TIP, which shows the “opposite” sign behavior. That thermal
vorticity leads to the “opposite” sign, as seen in many early studies
Becattini and Karpenko (2018); Xia  _et al._ (2018); Fu  _et al._ (2020), is
sometimes referred to as “spin sign puzzle”.

Given SIP contribution is comparable to TIP contribution in magnitude but is
qualitatively different in ”sign”, the competition between SIP and TIP will
eventually determine the azimuthal angle dependence of the total spin
polarization. This competition is best seen in Fig. 2, where the total spin
polarization as a function of $\phi_{p}$ are drastically different with and
without SIP. In “Lambda equilibrium” scenario, though, TIP wins the
competition and the total polarization is the “opposite” sign even in the
presence of SIP. However, SIP becomes more important when the mass of spin
carrier becomes smaller. This can be easily understood from Eqs. (Shear-
induced spin polarization in heavy-ion collisions),(5) that the factor
$|p^{2}_{\perp}|/\varepsilon^{2}_{0}$ (the square of typical velocity of
fermions) is larger when the spin carrier is lighter. Indeed, in the “strange
memory” scenario, SIP prevails over TIP in both $P_{z}$ and $P_{y}$. Seeing
this, we should not be surprised to find that the total spin polarization
shows the “same” sign.

To complement Figs. 1 2, we compare our results in the particle rest frame as
shown above with those in the lab frame in Fig. 3. This comparison illustrates
the sensitivity of spin polarization to the choice of the reference frame. We
notice that $P_{y}$ is more sensitive to the change of the reference frame
than $P_{z}$; see also Ref Liu  _et al._ (2020) for related discussion.

Figure 3: The comparison between strange quark and $\Lambda$ hyperon spin
polarization in lab and particle rest frame.

To investigate the generality of the results reported above, we repeat our
calculations by systematically varying inputs for the present hydrodynamic
model, including initial conditions, freezeout temperature, and the
parametrization of the shear viscosity, bulk viscosity and equation of state
(EoS), see our upcoming publication for more details. In all cases that we
have explored, SIP always induces the “same sign” contribution whereas TIP
contribution is the “opposite sign”. In “Lambda equilibrium” scenario, the
total spin polarization is always ”opposite” sign. This should be contrasted
with the results seen in the “strange memory” scenario. Except for some
specific cases when a unusually large $T$-gradient is caused by EoSs much
harder than the lattice EoS or a sharply-peaked bulk viscosity around the
freezeout region, the azimuthal angle dependence of spin polarization is
generically dominated by SIP and shows the qualitative agreement with the
data.

Summary and discussion.—In this letter, we investigate the shear-induced spin
polarization (SIP), which is an undiscovered mechanism for spin polarization
generation. We study the effects of SIP and thermal vorticity on the azimuthal
angle dependence of $\Lambda$ spin polarization in heavy-ion collisions at
$\sqrt{s_{NN}}=200$ GeV based on a data-calibrated hydrodynamic model. Our
main results, summarized in Fig. 1 and Fig. 2, clearly demonstrate that SIP
gives rise to distinctive new features in differential spin polarization and
is indispensable to analyze the effects induced by the hydrodynamic gradient
on the measured spin polarization. By contrasting the results in the “Lambda
equilibrium” scenario where the strange quark’s memory is completely forgotten
and those in the “strange memory” scenario, it is tempting to conclude that
the presence of both SIP and the memory of strange quarks is required for the
azimuthal angle dependence of spin polarization to agree with data
qualitatively.

For future quantitative studies, it is important to understand how to form a
polarized $\Lambda$ from the polarized quarks and the subsequent evolution in
the hadronic stage Li and Yee (2019); Wang  _et al._ (2020); Yang  _et al._
(2020); Weickgenannt  _et al._ (2020, 2021); Li and Liu (2020); Sheng  _et
al._ (2020); Wang and Zhuang (2021). Future studies should also investigate
the evolution of quark spin polarization in QGP Florkowski  _et al._ (2018);
Hattori  _et al._ (2019a); Fukushima and Pu (2020); Li  _et al._ (2020);
Bhadury  _et al._ (2021). We limit our calculations to high-energy heavy-ion
collisions. SIP should be present in collisions at the beam scan energies at
RHIC and could potentially be employed to explore the properties of QCD matter
at finite baryon density.

###### Acknowledgements.

We thank helpful discussions with Shanshan Cao, Xu Cao, Hengtong Ding, Fei
Gao, Feng Li, Yanf and Yifeng Sun. This work was supported in part by the NSFC
under grant No. 12075007 and No. 11675004 (BF and HS) and by No. 11861131009
and No. 12075098 (LP) as well as by the Strategic Priority Research Program of
Chinese Academy of Sciences, Grant No. XDB34000000 (SL and YY). We acknowledge
the extensive computing resources provided by the Supercomputing Center of
Chinese Academy of Science (SCCAS), Tianhe-1A from the National Supercomputing
Center in Tianjin, China and the High-performance Computing Platform of Peking
University as well as Nuclear Science Computer Center at CCNU (NSC3), China.

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