M

<EMAIL_ADDRESS>Kotla Rekis Bao Eisele Noohinejad
Tolkiehn Paulmann Singh Verma Bag Kulkarni Thamizhavel
<EMAIL_ADDRESS><EMAIL_ADDRESS>[a]Sandervan Smaalensmash@uni-
bayreuth.de

[a]Laboratory of Crystallography, University of Bayreuth, 95447 Bayreuth
Germany

[b]Department of Quantum Matter, AdSM, Hiroshima University, 739-8530,
Higashi-Hiroshima Japan

[c]Department of Physics, Materials Genome Institute and International Center
for Quantum and Molecular Structures, Shanghai University, Shanghai 200444, P.
R. China

[d]P24, PETRA III, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg Germany

[e]Mineralogisch-Petrographisches Institut,Universität Hamburg, 20146 Hamburg
Germany

[f]Department of Condensed Matter Physics and Materials Science,Tata Institute
of Fundamental Research, Mumbai 400005 India

# Orthorhombic charge density wave on the tetragonal lattice of EuAl4

Surya Rohith  Toms  Jin-Ke  Claudio  Leila  Martin  Carsten  Birender  Rahul
Biplab  Ruta  Arumugam

###### Abstract

EuAl4 possesses the BaAl4 crystal structure type with tetragonal symmetry
$I4/mmm$. It undergoes a charge-density-wave (CDW) transition at $T_{CDW}=145$
K and it features four consecutive antiferromagnetic phase transitions below
16 K. Here, we use single-crystal x-ray diffraction to determine
incommensurately modulated crystal structure of EuAl4 in its CDW state. The
CDW is shown to be incommensurate with modulation wave vector
$\mathbf{q}=(0,0,0.1781(3))$ at 70 K. The symmetry of the incommensurately
modulated crystal structure is orthorhombic with superspace group
$Fmmm(00\sigma)s00$, where $Fmmm$ is a subgroup of $I4/mmm$ of index 2. Both
the lattice and the atomic coordinates of the basic structure remain
tetragonal. Symmetry breaking is entirely due to the modulation wave, where
atoms Eu and Al1 have displacements exclusively along $\mathbf{a}$, while the
fourfold rotation would require equal displacement amplitudes along
$\mathbf{a}$ and $\mathbf{b}$. The calculated band structure of the basic
structure and interatomic distances in the modulated crystal structure both
indicate the aluminum atoms as location of the CDW. The temperature dependence
of the specific heat reveals an anomaly at $T_{CDW}=145$ K of a magnitude
similar to canonical CDW systems. The present discovery of orthorhombic
symmetry for the CDW state of EuAl4 leads to the suggestion of monoclinic
instead of orthorhombic symmetry for the third AFM state.

###### keywords:

Charge density wave, Twinning, Modulated, Superspace

The incommensurate charge-density-wave of EuAl4 below $T_{CDW}=145$ K is found
to possess orthorhombic symmetry, despite an average crystal structure that
remains tetragonal in very good approximation. This finding has ramifications
for the interpretation of all physical properties of EuAl4, in particular its
multiple magnetic transitions.

##  1 Introduction

EuAl4 has attracted attention, because it develops a charge-density wave (CDW)
below $T_{CDW}=145.1$ K and it exhibits four successive magnetic transitions
below 16 K [nakamura2015a, shimomura2019a]. EuAl4 adopts the BaAl4 structure
type that has symmetry according to the tetragonal space group $I4/mmm$
[parth1983a, nakamura2015a], as shown in Fig. 1. It belongs to a large family
of isostructural compounds, including magnetic EuGa4, fully ordered EuAl2Ga2
and non-magnetic SrAl4 and BaAl4 [nakamura2016a, stavinoham2018a, onuki2020a].
Recently, a symmetry-protected non-trivial topology of the electronic band
structure was proposed for BaAl4 [wangk2021a]. In case of magnetic EuAl4, a
chiral spin structure, like skyrmions reported in some divalent Eu compounds
such as EuPtSi [Kaneko2019a, onuki2020a], was proposed on the basis of the
observation of the topological Hall resistivity and muon-spin rotation and
relaxation ($\mu SR$) studies [shang2021a, zhuxy2022a]. If such a nontrivial
texture would be confirmed, EuAl4 would represent a rare case of a compound
where one could observe the coexistence of exotic magnetic order and a CDW.
Since these exotic electronic and spin structures depend on the symmetry,
knowledge of the true symmetry of the crystal structure thus is of utmost
importance for understanding non-trivial magnetic properties. Here we show
that the CDW transition of EuAl4 is accompanied by a lowering of the symmetry
towards orthorhombic, and we present the incommensurately modulated CDW
crystal structure.

EuAl4 is one of a few compounds [schuttewj1993b], where the lowering of the
crystal symmetry at a phase transition is governed by the symmetry of the
incommensurate modulation wave describing the CDW, while any lattice
distortion could not be detected in the present high-resolution diffraction
experiment with synchrotron radiation. This feature might explain why the
lowering of symmetry has not been found in earlier studies on EuAl4. This
behavior is in contrast to other CDW materials, like Er2Ir3Si5, Lu2Ir3Si5 and
BaFe2Al9, for which the CDW transitions are accompanied by large lattice
distortions [ramakrishnan2020a, ramakrishnan2021a, meierwr2021a]

The phenomenon of CDW was originally identified as a property of crystals with
quasi-one-dimensional (1D) electron bands, such as NbSe3 and K0.3MoO3
[grunerg1994, monceaup2012a]. A CDW is formed due to Fermi surface nesting
(FSN), where the nesting vector of the periodic structure becomes the wave
vector of the CDW of the metallic bands as well as of the accompanying
modulation of the atomic positions (periodic lattice distortion—PLD). The
modulation wave vector can be commensurate or incommensurate with respect to
the underlying periodic basic structure. More recent research has found that
CDWs can develop in crystalline materials that lack the 1D property of their
crystal structures and rather possess three-dimensionally (3D) structured
electron bands. Alternate mechanisms have been proposed for the formation of
CDWs in 3D compounds, including the mechanism of $q$-dependent electron-phonon
coupling (EPC) [zhux2015a, zhux2017a]. Strongly correlated electron systems
may also support the formation of CDWs [chencw2016a]. The latter are often
found for rare-earth containing intermetallic compounds, like the series of
isostructural compounds $R_{5}$Ir4Si10 ($R$ = rare earth) [ramakrishnan2017a].

The interplay between CDWs and magnetism in rare-earth compounds continues to
attract attention. A competition between these two symmetry-breaking phenomena
can be expected, because both CDWs and magnetic order depend on the Fermi
surface through FSN and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction
between the localized magnetic moments of the $4f$ electrons, as it is found
for EuAl4 [kobata2016a]. In case of magnetoelastic coupling, the lattice
distortion may couple to the PLD or the lattice distortion in the CDW state or
to the EPC. Experimentally, the coexistence of a CDW and antiferromagnetic
(AFM) order has been established for Er5Ir4Si10 and Sm2Ru3Ge5 [gallif2002a,
kuo2020a]. The series of compounds $R$NiC2 ($R$ = Pr, Nd, Gd, Tb, Dy, Ho, Er)
exhibit both CDW and AFM phase transitions [romanm2018a, shimomuras2016a,
kolinciokk2017a, maeda2019a]. SmNiC2 is an exception in this series, since it
develops ferromagnetic (FM) order below $T_{C}=17.7$ K at which transition the
CDW is destroyed [shimomura2009a, wolfel2010a]. Recently, coexistence of CDW
and FM orders was found for the field-induced FM state of TmNiC2
[kolincio2020a].

Magnetism of EuAl4 is related to localized magnetic moments of the europium
atoms in their divalent state: the electronic configuration $4f^{7}$ implies
$J=S=7/2$ and $L=0$, where $J$ is the total angular momentum, $S$ is the spin
angular momentum and $L$ is the orbital angular momentum [wernickjh1967a,
nakamura2015a]. This allows the study of the collective magnetism without
single-ion anisotropy, as divalent Eu has zero orbital angular momentum.
Magnetic interactions are governed by the RKKY interaction [nakamura2015a].
Neutron diffraction has established that the AFM order involves an
incommensurate modulation wave [kaneko2021a]. Single-crystal x-ray diffraction
(SXRD) has shown the coexistence of the incommensurate CDW modulation and AFM
order [shimomura2019a]. Furthermore, shimomura2019a proposed a lowering of the
lattice symmetry at the third magnetic transition towards $Immm$ orthorhombic.
This is essentially different from the present discovery of $Fmmm$
orthorhombic symmetry below the CDW transition.

EuAl4 possesses a 3D band structure with localized $4f$ electrons of Eu well
below the Fermi surface [kobata2016a]. The CDW mainly involves orbitals of the
Al atoms [kaneko2021a]. This is in agreement with the observation of a CDW
with $T_{CDW}$ = 243 K in isostructural SrAl4, where non-magnetic Sr replaces
Eu [nakamura2016a, haruo2020a]. Here we present electronic band-structure
calculations for the tetragonal structure of EuAl4. They confirm that the
location of the CDW is on the Al atoms. They reveal a 3D band structure with a
highly structured Fermi surface.

##  2 Experimental and Computational details

Single-crystals of EuAl4 were synthesized by the Al self-flux method. The
elements europium (Lieco, 99.9% purity) and aluminum (Alfa Aesar, 99.999%)
were filled into an alumina crucible in the ratio 1:20. The crucible was
sealed in an evacuated quartz glass ampoule. It was heated to a temperature of
1323 K and held at this temperature for 24 hours. After which the crucible was
cooled down to 1073 K in 6 hours and then slowly cooled at a rate of 2 K/hr
down to 923 K at which point the crystals were separated from the molten metal
by centrifugation. The 1:4 stoichiometry of the product was confirmed by
energy-dispersive x-ray spectroscopy (EDX) as well as by the structure
refinements against SXRD data.

X-ray diffraction experiments were performed at Beamline P24 of PETRA-III at
DESY in Hamburg, employing radiation of a wavelength of 0.5000 Å. The
temperature of the specimen was controlled by a CRYOCOOL open-flow helium gas
cryostat. Complete data sets of intensities of Bragg reflections were measured
at temperatures of 250 K (tetragonal phase), and of 70 K and 20 K (CDW phase).
Each run of data collection comprises $3640$ frames, corresponding to a
rotation of the crystal over 364 deg, which was repeated 10 times. These data
were binned to a data set of 364 frames of 1 deg of rotation and 10 seconds
exposure time, using the SNBL toolbox [dyadkinv2016a]. See Section S1 in the
supporting information.

The EVAL15 software suite [schreursamm2010a] has been used for processing the
SXRD data. At 250 K a single run was collected at a crystal-to-detector
distance of at 110 mm and without a $2\theta$ offset of the detector. At 70 K
and at 20 K a crystal-to-detector distance of 260 mm required two runs, with
and without $2\theta$ offset, respectively. The two binned runs for 70 K and
those for 20 K were integrated separately, and subsequently merged in the
module ANY of EVAL15. SADABS [sheldrick2008] has been used for scaling and
absorption correction with Laue symmetry $4/mmm$ for the 250 K data and $mmm$
for 70 K and 20 K. The reflection file produced was imported to Jana2006
[petricekv2014a, petricekv2016a]. Table 1 shows the crystallographic
information.

The magnetic susceptibility $\chi(T)$ has been measured for temperatures
2.5–300 K, using a commercial SQUID magnetometer (MPMS5 by Quantum Design,
USA). Measurements have been made in fields of 0.1 T and 0.5 T.

The specific heat, $C_{p}(T)$, has been measured from 220 to 8 K by the
thermal relaxation method, using a physical property measuring system (PPMS,
Quantum Design, USA).

Density functional theory (DFT) based calculations were performed within the
generalized gradient approximation (GGA) using the projector augmented (PAW)
wave method as implemented in the Vienna Ab-initio Simulation Package (VASP)
[Kresse1999a, Kresse1996a]. The Perdew-Burke-Ernzerhof (PBE) functional was
used to consider the exchange-correlation effects [Perdew1996a]. An energy
cutoff of 380 eV was used for the plane-wave basis set, and a
$\Gamma$-centered, $9\times 9\times 9$ $k$ mesh was employed for the bulk
Brillouin zone sampling. Spin-orbit coupling effects were considered in all
the calculations. We employed Eu2+ by considering the remaining $4f$ electrons
as core electrons. A tight-binding Hamiltonian was generated to compute the
Fermi surface on a finer $k$ grid [Marzari1996a]. The FermiSurfer software
package was used to visualize the Fermi surface [Kawamura2019a].

##  3 Discussion

###  3.1 Analysis of the CDW structure

SXRD at 250 K confirmed the $I4/mmm$ crystal structure of EuAl4. The SXRD data
at 70 K revealed satellite reflections at positions that can be described by
the modulation wave vector $\mathbf{q}=(0,0,0.1781(3))$, in agreement with the
results by nakamura2015a and shimomura2019a. Visualisation of the SXRD data
was done with aid of the software CrysAlisPRO [crysalis]. Figs. 2(a) and 2(b)
show a small part of the $(0,k,l)$ plane of the reconstructed reciprocal
lattice. Satellite reflections along $\mathbf{c}^{*}$ are clearly visible at
70 K [Fig. 2(b)]. Upon further cooling to 20 K, there is a reduction by 0.004
in the $\sigma_{3}$ component of the modulation wave vector, in agreement with
shimomura2019a.

We note that the lattice parameters in the CDW phase do not give evidence for
an orthorhombic lattice distortion. This could explain why earlier works have
not found this symmetry lowering.

In order to determine the crystal structure of the CDW phase, we have tested
different superspace groups for its symmetry (See Table S2 in the supporting
information). It is noticed that the tetragonal lattice allows two
fundamentally different orthorhombic lattices as subgroups: $Immm$ preserves
the mirror planes perpendicular to the $\mathbf{a}$ and $\mathbf{b}$ axes of
$I4/mmm$, while $Fmmm$ preserves the diagonal mirror planes. Table 2 provides
the crystallographic data for three of the refined crystal structures (compare
Table S2 in the supporting information). The three models, A, B and C, are
discussed below.

####  3.1.1 Model A:

In the diffraction pattern we did not observe any splitting of the main or
satellite reflections, where split reflections would indicate a twinned
crystal of lower symmetry. Also, we did not find a distortion in the lattice
parameters, as it would occur for a single-domain crystal of lower symmetry.
Furthermore, the preservation of tetragonal symmetry within the CDW phase was
reported in the literature [nakamura2015a, shimomura2019a]. Therefore, initial
data processing was performed under the assumption of tetragonal symmetry,
employing point group $4/mmm$ for scaling and absorption correction in SADABS
[sheldrick2008]). Structure refinements of the incommensurately modulated
structure were performed with a model with superspace group
$I4/mmm(00\sigma)0000$. Table 2 shows that $R_{int}$ as well as $R_{F}$ for
the main reflections are reasonably low, indicating that the average structure
of the CDW phase still is tetragonal in good approximation. This conclusion is
reinforced by the fact that refinement of the average structure against main
reflections leads to $R_{F}=1.55\,\%$; an excellent fit. However,
$R_{F}=60.46\,\%$ for the satellite reflections. This high value indicates
that the satellite reflections are not well fitted and that the CDW modulation
does not have tetragonal symmetry.

This makes Model A an unsuitable candidate for the incommensurate CDW
structure. Other tetragonal superspace groups were also tested, leading to
similar failures in describing the modulation wave or with reflection
conditions violated by the measured SXRD data (see Table S2 in the supporting
information). As a result we can rule out tetragonal symmetry for the
modulated CDW crystal structure.

####  3.1.2 Model B:

As second model we have considered a lowering of the symmetry from tetragonal
$I4/mmm$ to its orthorhombic subgroup $Immm$. This orthorhombic point symmetry
was used for scaling and absorption correction of the SXRD data in SADABS
[sheldrick2008]). The CDW phase transition allows for pseudomerohedral
twinning of two, differently oriented domains on the tetragonal lattice, that
are related by the missing fourfold rotation [parsons2003a]. Since split
reflections or a lattice distortion could not be detected in the SXRD data,
all Bragg reflections have contributions from both domains. The structure
refinement of a model in superspace group $Immm(00\sigma)s00$ has lead to a
twin volume ratio of $0.485:0.515$, thus explaining the nearly tetragonal
point symmetry of the SXRD data. $R$-values indicate a good fit to the SXRD
data for this model (Table 2). As a result, this model is a prime candidate
for describing the incommensurately modulated crystal structure of the CDW
phase.

####  3.1.3 Model C:

As last model we present model C with symmetry according to $Fmmm$, the other
orthorhombic subgroup, which now preserves the diagonal mirror planes of
$I4/mmm$. Scaling and absorption correction of the SXRD data was performed
with SADABS according to the differently oriented point group $mmm$
[sheldrick2008]. Again, two domains are possible that are related by the
missing fourfold rotation. The structure refinement of a model in superspace
group $Fmmm(00\sigma)s00$ has lead to a twin volume ratio of
$0.454(4)\>:\>0.546$, thus explaining the nearly tetragonal point symmetry of
the SXRD data. $R$-values indicate an excellent fit to the SXRD data for this
model (Table 2), which is significantly better than that of model B.
Furthermore, the refined parameters possess slightly smaller s.u.’s in model C
than in model B, while the number of parameters is one smaller in model C
(Tables S3 and S4 in the supporting information). Therefore, the best fit to
the SXRD data has been obtained for a modulated crystal structure with
symmetry according to the superspace group $Fmmm(00\sigma)s00$.
$Fmmm(00\sigma)0s0$ is an alternate setting of this superspace group, while
all other symmetries lead to a worse fit to the SXRD data (Table S2 in the
supporting information).

Recently, kaneko2021a have proposed that the CDW of EuAl4 involves
displacements of the Al atoms perpendicular to $\mathbf{c}$, while Eu would
not be involved in the PLD. The present crystal structure involves atomic
modulations exclusively perpendicular to $\mathbf{c}$, as it is enforced by
the superspace symmetry (Table S4). However, the modulation amplitudes are of
comparable magnitude for all three atoms, Eu, Al1 and Al2. Nevertheless, a
non-zero modulation amplitude is not evidence by itself, that the involved
atom must contribute electronic states to the CDW. The atomic modulation may
also be caused by the elastic coupling to other atoms that are carrying the
CDW. In EuAl4, the shortest interatomic distances are between Al2 atoms and
for Al1–Al2 [Fig. 3(a)]. They are hardly modulated, and forming a two-
dimensional network of Al perpendicular to $\mathbf{c}$ (Fig. 1). The largest
modulation is found for the next shorter Al–Al distance between Al1 atoms
[Fig. 3(b)]. This strong modulation suggests that the CDW resides on the
layers of Al atoms. Eu is elastically coupled to Al1 and Al2 (Fig. S1) and is
not part of the CDW. The $t$ plots of interatomic distances cannot elaborate
on the precise location: the CDW resides either on the Al1 atoms or on a
network of Al1 and Al2 atoms (Fig. 3).

###  3.2 Electronic structure and Fermi surface

Figure 4(a) shows the calculated band structure along the high-symmetry
directions in the primitive Brillouin zone of the periodic crystal structure
of EuAl4 with $I4/mmm$ symmetry. Both the valence and conduction bands cross
the Fermi level $E_{F}$, resolving its metallic ground state. Importantly, the
bands along $\Gamma$–$Z$ have a substantial energy dispersion. This indicates
the three-dimensional nature of the Fermi surface as discussed below. There is
a Dirac nodal crossing above the Fermi level along the $\Gamma$–$Z$ direction,
which is protected by $C_{4z}$ rotational symmetry. EuAl4 thus realizes a
Dirac semimetallic state. To resolve the electronic states near $E_{F}$, we
present the atom-projected density of states (PDOS) in Fig. 4(b); Eu PDOS is
in blue and Al PDOS is in red. The Al states are dominant at $E_{F}$,
indicating that Al atoms are predominantly metallic and more likely undergo
CDW modulations, in agreement with the analysis of PLD (Section 3.1) and the
literature. PDOS of Eu comprises of $d$ states at the Fermi level, while $4f$
states are well below $E_{F}$, in agreement with the literature [kobata2016a].

We present the calculated Fermi pockets associated with the valence ($h^{+}$)
and conduction states ($e^{-}$) in Figs. 4(c,d). They reveal a hole pocket
centered on $\Gamma$ and an electron Fermi pocket centered on $Z$. Both Fermi
pockets are highly structured. The $e^{-}$ Fermi pocket suggest the
possibility of nesting perpendicular to the $\mathbf{c}$ axes. For the $h^{+}$
pocket a possible FSN is not clearly resolved. It should however be noted that
because of the 3D nature of the Fermi surface, the nesting may have a
complicated structure. These results do not clearly indicate FSN as mechanism
for the formation of the CDW state in EuAl4.

###  3.3 Magnetic susceptibility

The temperature dependence of the magnetic susceptibility, measured with
magnetic fields of 0.1 T and 0.5 T, is shown for 2.4–300 K in Fig. 5. Any
change of the susceptibility at the CDW transition (e.g. as a change of Pauli
paramagnetism) is masked by the large value of the paramagnetic
susceptibility. However, the low temperature data reveal an AFM transition
below 16 K, which agrees with the previously published value [nakamura2015a].

###  3.4 Specific heat

The temperature dependence of the specific heat ($C_{p}$) is shown for 8–210 K
in Fig. 6. The high temperature data clearly reveal a small, broad jump of
$\Delta C_{p}$ = 2.5 J/(mol K) at 145 K, suggesting a thermodynamic phase
transition (CDW) at 145 K. Such an anomaly in the temperature dependence of
the specific heat is consistent with that observed in canonical CDW systems,
like NbSe3 [tomi1981a]. The low-temperature data display multiple AFM
transitions, which have also been observed in earlier studies [nakamura2015a].

##  4 Conclusions

EuAl4 possesses the BaAl4 crystal structure type with tetragonal symmetry
$I4/mmm$. It undergoes a CDW transition at $T_{CDW}=145$ K. Here, we have
presented the incommensurately modulated crystal structure of EuAl4 in its CDW
state. Structure refinements according to the superspace approach have shown
that: (i) the modulation is incommensurate with modulation wave vector
$\mathbf{q}=(0,0,0.1781(3))$ at 70 K, in agreement with shimomura2019a; and
(ii) the symmetry of the CDW crystal structure is orthorhombic with superspace
group $Fmmm(00\sigma)s00$, where $Fmmm$ is a subgroup of $I4/mmm$ of index 2.
Despite this group–subgroup relation, we did not find any lattice distortion
in the SXRD data. Even more, atomic positions of the basic structure of
$Fmmm(00\sigma)s00$ still obey the $I4/mmm$ symmetry (Table S3). Symmetry
breaking is entirely in the modulation wave (CDW and PLD), where atoms Eu and
Al1 have displacements exclusively along $\mathbf{a}$ of the $F$-centered unit
cell (Table S4), while the fourfold rotation would require equal displacement
amplitudes along $\mathbf{a}$ and $\mathbf{b}$.

One interesting question is the location of the CDW. Analysis of the
modulation of interatomic distances (Section 3.1) as well as features of the
electronic band structure (Section 3.2) have indicated the Al atoms as
supporting the CDW, in agreement with the literature [kobata2016a,
kaneko2021a].

Compounds of rare earth ($R$) and transition metals, like $R_{5}$Ir4Si10 and
$R_{2}$Ir3Si5, contain highly correlated electron systems, with accompanying
influence on the mechanism of formation of CDWs. In EuAl4, the majority
element is the light $p$-block metal aluminum. Band-structure calculations
have shown that FSN is a possible mechanism of CDW formation. Furthermore, the
weak anomaly in $C_{p}(T)$ near $T_{CDW}$ is similar to anomalies of canonical
CDW materials and it is much smaller than observed for the rare-
earth–transition-metal base compounds, again this would support the FSN
mechanism. An alternative possibility is that the CDW is related to nesting of
nontrivial bands that are present in the band structure [shiw2021a,
chiuw2022a].

The present discovery of orthorhombic symmetry for the CDW state of EuAl4 is
important for modeling of the electronic properties of the CDW state as well
as for identifying the correct magnetic order and understanding the magnetic
properties of the four AFM states below 16 K.

Recent work either has used tetragonal symmetry for analysing the AFM states
[shang2021a, kaneko2021a]. Alternatively, shimomura2019a have proposed
orthorhombic $Immm$ symmetry for the third AFM state. This orthorhombic
subgroup incorporates the perpendicular mirror planes of $I4/mmm$, while
presently found $Fmmm$ is based on the diagonal mirror planes of $I4/mmm$. On
the other hand, shimomura2019a report peak splitting in neutron diffraction
into ”three maxima.” Together with the present observation of lowering of
symmetry at the CDW transition, this suggests that the third AFM state could
have monoclinic symmetry ($\mathbf{c}$ unique) instead of orthorhombic
symmetry, since, apparently, both the diagonal and perpendicular mirror planes
are lost.

Acknowledgements We acknowledge DESY (Hamburg, Germany), a member of the
Helmholtz Association HGF, for the provision of experimental facilities. Parts
of this research were carried out at PETRA III, using beamline P24. Beamtime
was allocated for proposal I-20190810. J.‐K. Bao acknowledges the Alexander
von Humboldt Foundation for financial support.

## References

  * [1] [Chen _et al._]Chen, Choe Morosan2016chencw2016a Chen, C.-W., Choe, J. Morosan, E. 2016.  _Rep. Prog. Phys._ 79(8), 084505.
  * [2] [Chiu _et al._]Chiu, Chang, Macam, Belopolski, Huang, Markiewicz, Yin, jia Cheng, Lee, Chang, Chuang, Xu, Lin, Hasan Bansil2022chiuw2022a Chiu, W.-C., Chang, G., Macam, G., Belopolski, I., Huang, S.-M., Markiewicz, R., Yin, J.-X., jia Cheng, Z., Lee, C.-C., Chang, T.-R., Chuang, F.-C., Xu, S.-Y., Lin, H., Hasan, M. Z. Bansil, A. 2022.  _ArXiv:_ , 2201.04757.
  * [3] [Dyadkin _et al._]Dyadkin, Pattison, Dmitriev Chernyshov2016dyadkinv2016a Dyadkin, V., Pattison, P., Dmitriev, V. Chernyshov, D. 2016\.  _J. Synchrotron Rad._ 23(3), 825–829.
  * [4] [Galli _et al._]Galli, Feyerherm, Hendrikx, Dudzik, Nieuwenhuys, Ramakrishnan, Brown, van Smaalen Mydosh2002gallif2002a Galli, F., Feyerherm, R., Hendrikx, R. W. A., Dudzik, E., Nieuwenhuys, G. J., Ramakrishnan, S., Brown, S. D., van Smaalen, S. Mydosh, J. A. 2002\.  _J. Phys.: Condens. Matter_ , 14, 5067–5075.
  * [5] Gruner1994grunerg1994 Gruner, G. 1994.  _Density waves in solids_.  Reading, Masachusetts: Addison–Wesley.
  * [6] [Kaneko _et al._]Kaneko, Frontzek, Matsuda, Nakao, Munakata, Ohhara, Kakihana, Haga, Hedo, Nakama Ōnuki2019Kaneko2019a Kaneko, K., Frontzek, M. D., Matsuda, M., Nakao, A., Munakata, K., Ohhara, T., Kakihana, M., Haga, Y., Hedo, M., Nakama, T. Ōnuki, Y. 2019\.  _J. Phys. Soc. Jpn._ 88(1), 013702.
  * [7] [Kaneko _et al._]Kaneko, Kawasaki, Nakamura, Munakata, Nakao, Hanashima, Kiyanagi, Ohhara, Hedo, Nakama Onuki2021kaneko2021a Kaneko, K., Kawasaki, T., Nakamura, A., Munakata, K., Nakao, A., Hanashima, T., Kiyanagi, R., Ohhara, T., Hedo, M., Nakama, T. Onuki, Y. 2021\.  _J. Phys. Soc. Jpn._ 90(6), 064704.
  * [8] Kawamura2019Kawamura2019a Kawamura, M. 2019.  _Comput. Phys. Commun._ 239, 197.
  * [9] [Kobata _et al._]Kobata, Fujimori, Takeda, Okane, Saitoh, Kobayashi, Yamagami, Nakamura, Hedo, Nakama Onuki2016kobata2016a Kobata, M., Fujimori, S.-i., Takeda, Y., Okane, T., Saitoh, Y., Kobayashi, K., Yamagami, H., Nakamura, A., Hedo, M., Nakama, T. Onuki, Y. 2016\.  _J. Phys. Soc. Jpn._ 85(9), 094703.
  * [10] [Kolincio _et al._]Kolincio, Roman Klimczuk2020kolincio2020a Kolincio, K. K., Roman, M. Klimczuk, T. 2020.  _Phys. Rev. Lett._ 125, 176601.
  * [11] [Kolincio _et al._]Kolincio, Roman, Winiarski, Strychalska-Nowak Klimczuk2017kolinciokk2017a Kolincio, K. K., Roman, M., Winiarski, M. J., Strychalska-Nowak, J. Klimczuk, T. 2017.  _Phys. Rev. B_ , 95(23), 235156.
  * [12] Kresse Furthmüller1996Kresse1996a Kresse, G. Furthmüller, J. 1996.  _Phys. Rev. B_ , 54, 11169.
  * [13] Kresse Joubert1999Kresse1999a Kresse, G. Joubert, D. 1999.  _Phys. Rev. B_ , 59, 1758.
  * [14] [Kuo _et al._]Kuo, Hsu, Tseng, Chen, Lin, Liu, Kuo Lue2020kuo2020a Kuo, C. N., Hsu, C. J., Tseng, C. W., Chen, W. T., Lin, S. Y., Liu, W. Z., Kuo, Y. K. Lue, C. S. 2020.  _Phys. Rev. B_ , 101, 155140.
  * [15] [Maeda _et al._]Maeda, Kondo Nogami2019maeda2019a Maeda, H., Kondo, R. Nogami, Y. 2019.  _Phys. Rev. B_ , 100, 104107.
  * [16] Marzari Vanderbilt1997Marzari1996a Marzari, N. Vanderbilt, D. 1997.  _Phys. Rev. B_ , 56, 12847.
  * [17] [Meier _et al._]Meier, Chakoumakos, Okamoto, McGuire, Hermann, Samolyuk, Gao, Zhang, Stone, Christianson Sales2021meierwr2021a Meier, W. R., Chakoumakos, B. C., Okamoto, S., McGuire, M. A., Hermann, R. P., Samolyuk, G. D., Gao, S., Zhang, Q., Stone, M. B., Christianson, A. D. Sales, B. C. 2021.  _Chem. Mater._ 33, 2855–2863.
  * [18] Monceau2012monceaup2012a Monceau, P. 2012.  _Adv. Phys._ 61(4), 325–581.
  * [19] [Nakamura _et al._]Nakamura, Uejo, Harima, Araki, Kobayashi, Nakashima, Amako, Hedo, Nakama Onuki2016nakamura2016a Nakamura, A., Uejo, T., Harima, H., Araki, S., Kobayashi, T. C., Nakashima, M., Amako, Y., Hedo, M., Nakama, T. Onuki, Y. 2016.  _J. Alloys Compd._ 654, 290–299.
  * [20] [Nakamura _et al._]Nakamura, Uejo, Honda, Takeuchi, Harima, Yamamoto, Haga, Matsubayashi, Uwatoko, Hedo, Nakama Onuki2015nakamura2015a Nakamura, A., Uejo, T., Honda, F., Takeuchi, T., Harima, H., Yamamoto, E., Haga, Y., Matsubayashi, K., Uwatoko, Y., Hedo, M., Nakama, T. Onuki, Y. 2015.  _J. Phys. Soc. Jpn._ 84(12), 124711.
  * [21] [Niki _et al._]Niki, Kuroshima, Higa, Morishima, Yogi, Nakamura, Niki, Maehira, Hedo, Nakama Onuki2020haruo2020a Niki, H., Kuroshima, H., Higa, N., Morishima, M., Yogi, M., Nakamura, A., Niki, K., Maehira, T., Hedo, M., Nakama, T. Onuki, Y. 2020\.  _Proceedings of the International Conference on Strongly Correlated Electron Systems (SCES2019)_.
  * [22] [Ōnuki _et al._]Ōnuki, Hedo Honda2020onuki2020a Ōnuki, Y., Hedo, M. Honda, F. 2020.  _J. Phys. Soc. Jpn._ 89(10), 102001.
  * [23] Parsons2003parsons2003a Parsons, S. 2003.  _Acta Crystallogr. D_ , 59(11), 1995–2003.
  * [24] [Parthé _et al._]Parthé, Chabot, Braun Engel1983parth1983a Parthé, E., Chabot, B., Braun, H. F. Engel, N. 1983\.  _Acta Crystallogr. B_ , 39(5), 588–595.
  * [25] [Perdew _et al._]Perdew, Burke Ernzerhof1996Perdew1996a Perdew, J. P., Burke, K. Ernzerhof, M. 1996.  _Phys. Rev. Lett._ 77, 3865.
  * [26] [Petricek _et al._]Petricek, Dusek Palatinus2014petricekv2014a Petricek, V., Dusek, M. Palatinus, L. 2014.  _Z. Kristallogr._ 229, 345–352.
  * [27] [Petricek _et al._]Petricek, Eigner, Dusek Cejchan2016petricekv2016a Petricek, V., Eigner, V., Dusek, M. Cejchan, A. 2016.  _Z. Kristallogr._ 231(5), 301–312.
  * [28] [Ramakrishnan _et al._]Ramakrishnan, Schönleber, Bao, Rekis, Kotla, Schaller, van Smaalen, Noohinejad, Tolkiehn, Paulmann, Sangeetha, Pal, Thamizhavel Ramakrishnan2021ramakrishnan2021a Ramakrishnan, S., Schönleber, A., Bao, J.-K., Rekis, T., Kotla, S. R., Schaller, A. M., van Smaalen, S., Noohinejad, L., Tolkiehn, M., Paulmann, C., Sangeetha, N. S., Pal, D., Thamizhavel, A. Ramakrishnan, S. 2021\.  _Phys. Rev. B_ , 104, 054116.
  * [29] [Ramakrishnan _et al._]Ramakrishnan, Schönleber, Rekis, van Well, Noohinejad, van Smaalen, Tolkiehn, Paulmann, Bag, Thamizhavel, Pal Ramakrishnan2020ramakrishnan2020a Ramakrishnan, S., Schönleber, A., Rekis, T., van Well, N., Noohinejad, L., van Smaalen, S., Tolkiehn, M., Paulmann, C., Bag, B., Thamizhavel, A., Pal, D. Ramakrishnan, S. 2020.  _Phys. Rev. B_ , 101, 060101(R).
  * [30] Ramakrishnan van Smaalen2017ramakrishnan2017a Ramakrishnan, S. van Smaalen, S. 2017.  _Rep. Prog. Phys._ 80, 116501.
  * [31] Rigaku2019crysalis Rigaku, 2019.  Crysalis pro version 171.40.53, rigaku oxford diffraction.
  * [32] [Roman _et al._]Roman, Strychalska-Nowak, Klimczuk Kolincio2018romanm2018a Roman, M., Strychalska-Nowak, J., Klimczuk, T. Kolincio, K. K. 2018\.  _Phys. Rev. B_ , 97(4), 041103(R).
  * [33] [Schreurs _et al._]Schreurs, Xian Kroon-Batenburg2010schreursamm2010a Schreurs, A. M. M., Xian, X. Kroon-Batenburg, L. M. J. 2010\.  _J. Appl. Crystallogr._ 43, 70–82.
  * [34] [Schutte _et al._]Schutte, Disselborg de Boer1993schuttewj1993b Schutte, W. J., Disselborg, F. de Boer, J. L. 1993.  _Acta Crystallogr. B_ , 49, 787–794.
  * [35] [Shang _et al._]Shang, Xu, Gawryluk, Ma, Shiroka, Shi Pomjakushina2021shang2021a Shang, T., Xu, Y., Gawryluk, D. J., Ma, J. Z., Shiroka, T., Shi, M. Pomjakushina, E. 2021.  _Phys. Rev. B_ , 103, L020405.
  * [36] Sheldrick2008sheldrick2008 Sheldrick, G. M. 2008.  _SADABS, Version 2008/1_.  Göttingen: University of Göttingen.
  * [37] [Shi _et al._]Shi, Wieder, Meyerheim, Sun, Zhang, Li, Shen, Qi, Yang, Jena, Werner, Koepernik, Parkin, Chen, Felser, Bernevig Wang2021shiw2021a Shi, W., Wieder, B. J., Meyerheim, H. L., Sun, Y., Zhang, Y., Li, Y., Shen, L., Qi, Y., Yang, L., Jena, J., Werner, P., Koepernik, K., Parkin, S., Chen, Y., Felser, C., Bernevig, B. A. Wang, Z. 2021.  _Nat. Phys._ 17(3), 381–387.
  * [38] [Shimomura _et al._]Shimomura, Hayashi, Asaka, Wakabayashi, Mizumaki Onodera2009shimomura2009a Shimomura, S., Hayashi, C., Asaka, G., Wakabayashi, N., Mizumaki, M. Onodera, H. 2009.  _Phys. Rev. Lett._ 102, 076404.
  * [39] [Shimomura _et al._]Shimomura, Hayashi, Hanasaki, Ohnuma, Kobayashi, Nakao, Mizumaki Onodera2016shimomuras2016a Shimomura, S., Hayashi, C., Hanasaki, N., Ohnuma, K., Kobayashi, Y., Nakao, H., Mizumaki, M. Onodera, H. 2016.  _Phys. Rev. B_ , 93, 165108.
  * [40] [Shimomura _et al._]Shimomura, Murao, Tsutsui, Nakao, Nakamura, Hedo, Nakama Onuki2019shimomura2019a Shimomura, S., Murao, H., Tsutsui, S., Nakao, H., Nakamura, A., Hedo, M., Nakama, T. Onuki, Y. 2019.  _J. Phys. Soc. Jpn._ 88(1), 014602.
  * [41] [Stavinoha _et al._]Stavinoha, Cooley, Minasian, McQueen, Kauzlarich, Huang Morosan2018stavinoham2018a Stavinoha, M., Cooley, J. A., Minasian, S. G., McQueen, T. M., Kauzlarich, S. M., Huang, C. L. Morosan, E. 2018.  _Phys. Rev. B_ , 97(19), 195146.
  * [42] [Stokes _et al._]Stokes, Campbell van Smaalen2011stokesht2011a Stokes, H. T., Campbell, B. J. van Smaalen, S. 2011.  _Acta Crystallogr. A_ , 67, 45–55.
  * [43] [Tomić _et al._]Tomić, Biljaković, Djurek, Cooper, Monceau Meerschaut1981tomi1981a Tomić, S., Biljaković, K., Djurek, D., Cooper, J., Monceau, P. Meerschaut, A. 1981.  _Solid State Commun._ 38, 109–112.
  * [44] [Wang _et al._]Wang, Mori, Wang, Wang, Ma, Latzke, Graf, Denlinger, Campbell, Bernevig, Lanzara Paglione2021wangk2021a Wang, K., Mori, R., Wang, Z., Wang, L., Ma, J. H. S., Latzke, D. W., Graf, D. E., Denlinger, J. D., Campbell, D., Bernevig, B. A., Lanzara, A. Paglione, J. 2021.  _npj Quantum Mater._ 6, 28.
  * [45] [Wernick _et al._]Wernick, Williams Gossard1967wernickjh1967a Wernick, J. H., Williams, H. J. Gossard, A. C. 1967.  _J. Phys. Chem. Solids_ , 28(2), 271–273.
  * [46] [Wölfel _et al._]Wölfel, Li, Shimomura, Onodera van Smaalen2010wolfel2010a Wölfel, A., Li, L., Shimomura, S., Onodera, H. van Smaalen, S. 2010\.  _Phys. Rev. B_ , 82, 054120.
  * [47] [Zhu _et al._]Zhu, Cao, Zhang, Plummer Guo2015zhux2015a Zhu, X., Cao, Y., Zhang, J., Plummer, E. W. Guo, J. 2015\.  _Proc. Natl. Acad. Sci. U. S. A._ 112(8), 2367–2371.
  * [48] [Zhu _et al._]Zhu, Guo, Zhang Plummer2017zhux2017a Zhu, X., Guo, J., Zhang, J. Plummer, E. W. 2017.  _Adv. Phys. X_ , 2(3), 622–640.
  * [49] [Zhu _et al._]Zhu, Zhang, Gawryluk, Zhen, Yu, Ju, Xie, Jiang, Cheng, Xu, Shi, Pomjakushina, Zhan, Shiroka Shang2022zhuxy2022a Zhu, X. Y., Zhang, H., Gawryluk, D. J., Zhen, Z. X., Yu, B. C., Ju, S. L., Xie, W., Jiang, D. M., Cheng, W. J., Xu, Y., Shi, M., Pomjakushina, E., Zhan, Q. F., Shiroka, T. Shang, T. 2022.  _Physical Review B_ , 105(1), 014423.
  * [50]

Table 1: Crystallographic data of crystal A of EuAl4 at 250 K, 70 K and 20 K. Refinement method used: Least-squares on $F$. The superspace group (SSG; No.) is given according to stokesht2011a. Temperature (K) | 250 | 70 | 20
---|---|---|---
Crystal system | Tetragonal | Orthorhombic | Orthorhombic
Space group; SSG | $I4/mmm$ | $Fmmm(00\sigma)s00$ | $Fmmm(00\sigma)s00$
No. | 139 | 69.1.17.2 | 69.1.17.2
$a$ (Å) | 4.3949(1) | 6.1992(4) | 6.1991(3)
$b$ (Å) | 4.3949 | 6.2001(4) | 6.1987(4)
$c$ (Å) | 11.1607(3) | 11.1477(3) | 11.1488(4)
Volume (Å3) | 215.57(1) | 428.47(4) | 428.41(4)
Wavevector, $q_{z}$ | - | 0.1781(3) | 0.1741(2)
$Z$ | 2 | 4 | 4
Wavelength (Å) | 0.50000 | 0.50000 | 0.50000
Detector distance (mm) | 110 | 260 | 260
$2\theta$-offset (deg) | 0 | 0, 25 | 0, 25
$\chi$-offset (deg) | -60 | -60 | -60
Rotation per image (deg) | 1 | 1 | 1
$(\sin(\theta)/\lambda)_{max}$ (Å-1) | 0.682610 | 0.748910 | 0.749031
Absorption, $\mu$ (mm-1) | 5.8373 | 5.9090 | 5.9100
Tmin, Tmax | 0.3211, 0.3712 | 0.3209, 0.3732 | 0.3192, 0.3676
Criterion of observability | $I>3\sigma(I)$ | $I>3\sigma(I)$ | $I>3\sigma(I)$
Number of main reflections |  |  |
measured | 1407 | 473 | 470
unique (obs/all) | 109/109 | 174/174 | 176/176
Number of satellites |  |  |
measured | - | 929 | 928
unique (obs/all) | - | 279/316 | 263/322
$R_{int}$ main (obs/all) | 0.0374/0.0374 | 0.0136/0.0136 | 0.0188/0.0188
$R_{int}$ sat (obs/all) | - | 0.0581/0.0588 | 0.0606/0.0616
No. of parameters | 9 | 18 | 18
$R_{F}$ main (obs) | 0.0147 | 0.0165 | 0.0213
$R_{F}$ sat (obs) | - | 0.0369 | 0.0311
$wR_{F}$ main (all) | 0.0214 | 0.0203 | 0.0230
$wR_{F}$ sat (all) | - | 0.0395 | 0.0336
$wR_{F}$ all (all) | 0.0214 | 0.0245 | 0.0250
GoF (obs/all) | 1.53/1.53 | 1.13/1.09 | 0.93/0.88
$\Delta\rho_{min}$, $\Delta\rho_{max}$(e Å-3) | -1.35, 1.15 | -2.40, 3.58 | -1.49, 1.58
Table 2: Crystallographic data for three models for the modulated crystal structure at 70 K, based on different superspace groups. Criterion of observability: $I>3\sigma(I)$ Model | A | B | C |
---|---|---|---|---
$a$ (Å) | 4.3834(3) | 4.3835(3) | 6.1992(4) |
$b$ (Å) | 4.3834 | 4.3841(3) | 6.2001(4) |
$c$ (Å) | 11.1488(4) | 11.1475(3) | 11.1477(3) |
$V$ (Å3) | 214.21(2) | 214.23(2) | 428.47(4) |
q | 0.1782(3)c* | 0.1781(3)c* | 0.1781(3)c* |
SSG | $I4/mmm(00\sigma)0000$ | $Immm(00\sigma)s00$ | $Fmmm(00\sigma)s00$ |
Rint main (obs/all)$\%$ | 1.53/1.53 | 1.28/1.28 | 1.36/1.36 |
Rint sat (obs/all)$\%$ | 7.43/7.49 | 6.75/6.85 | 5.81/5.88 |
RF main (obs/all)$\%$ | 4.37/4.37 | 1.96/1.96 | 1.65/1.65 |
RF sat (obs/all)$\%$ | 60.46/70.53 | 5.25/5.83 | 3.69/4.05 |
Unique main (obs/all) | 130/130 | 225/225 | 174/174 |
Unique sat (obs/all) | 215/254 | 365/425 | 279/316 |
No. of parameters | 13 | 19 | 18 |
Figure 1: Crystal structure of EuAl4 with space group $I4/mmm$ in the periodic
phase at 250K. Depicted is the $I$-centered unit cell with basis vectors
$\mathbf{a}_{I}$, $\mathbf{b}_{I}$ and $\mathbf{c}_{I}$. Brown spheres
correspond to the Eu atoms; dark blue spheres represent Al1 atoms; and green
spheres stand for Al2 atoms. Shortest interatomic distances are: $d$[Eu–Eu] =
4.3949(2) Å, $d$[Al1–Al1] = 3.1077(1) Å, $d$[Al2–Al1] = 2.664(1) Å and
$d$[Al2–Al2] = 2.568(4) Å. Figure 2: Excerpt of the reconstructed reciprocal
layer $(0\,k\,l)$ for SXRD data measured at (a) $T=250$ K, and (b) $70$ K.
Indices are given for several main reflections. Panel (b) is better resolved
than panel (a), because of the longer crystal-to-detector distance at 70 K.
Dark bands are due to insensitive pixels between the active modules of the
PILATUS3 X CdTe 1M detector.

Figure 3: $t$-Plot of interatomic distances (Å) $d$[Al1–Al1], $d$[Al2–Al1] and
$d$[Al2–Al2] at 70 K, where the first atom is the central atom. The number on
each curve is the number of the symmetry operator that is applied to the
second atom of the bond pair. Symmetry operators are listed in Table S5 in the
supporting information. Figure 4: (a) Bulk band structure and (b) Density-of-
states (DOS) of EuAl4. The dashed lines in (a) and (b) mark the Fermi level at
energy ($E$) zero. The calculated (c) hole (blue) and (d) electron (yellow)
Fermi pockets in the primitive bulk Brillouin zone. Three-dimensionally (3D)
structured hole and electron Fermi pockets are resolved.  Figure 5:
Temperature dependent magnetic susceptibility of EuAl4 from 2.4 to 300 K. Data
measured in fields of 0.1 T and 0.5 T. Figure 6: Temperature dependence of the
specific heat $C_{p}$ from 8 to 210 K. The lower inset (a) provides an
enlarged view around the anomaly at 145 K, where $\Delta C$ = 2.5 J/(mol K).
The upper inset (b) displays $C_{p}$ vs $T$ at low temperatures 8–25 K.


# Merger Effects on the Spin and Shape Alignments of Galaxy Stellar, Cold and
Hot Gas, and Dark Matter Components

Jounghun Lee and Jun-Sung Moon<EMAIL_ADDRESS>Astronomy program,
Department of Physics and Astronomy, Seoul National University, Seoul 08826,
Republic of Korea

###### Abstract

We present a numerical evidence supporting the scenario that the peculiar
alignments of the galaxy stellar spins with the major principal axes of the
local tidal tensors are produced during the quiescent evolution period when
the galaxies experience no recent merger events. Analyzing the merger tree
from the TNG300-1 simulation of the IllustrisTNG project, we find the latest
merger epochs, $a(z_{m})$, of the galaxies, and create four
$a(z_{m})$-selected samples that are controlled to share the identical mass
and density distributions. For each sample, we determine the spin and shape
vectors of the galaxy stellar, cold and hot gas, and dark matter components
separately, and compute the average strengths of their alignments with the
principal directions of the local tidal fields as well as their mutual
alignment tendencies. It is found that the stellar (cold gas) spin axes of the
galaxies whose latest merger events occur at earlier epochs are more strongly
aligned (weakly anti-aligned) with the major principal axes of the tidal
fields. It is also shown that although the mass-dependent transition of the
galaxy DM spins have little connection with the merger events, the
morphologies, spin-shape and shape-shear alignment strengths of the galaxy
four components sensitively depend on $a(z_{m})$. Noting that the stellar
components of the galaxies which undergo long quiescent evolution have
distinctively oblate shapes and very strong spin-shape alignments, we suggest
that the local tidal field might be traced by using the stellar shapes of
galaxies without signatures of mergers as a proxy of their stellar spins.

Unified Astronomy Thesaurus concepts: Large-scale structure of the universe
(902)

##  1 Introduction

The peculiar tidal connection of the galaxy stellar spins is a term first
coined by Lee, Moon & Yoon (2022) to describe the phenomenon detected in a
hydrodynamical simulation that the spin axes of the galaxy stellar components
tend to be aligned with the major principal directions of the local tidal
fields. This phenomenon is peculiar since the non-stellar gas and dark matter
(DM) components of the galaxies strongly prefer the directions perpendicular
to the major principal axes of the local tidal tensors in their spin
orientations, regardless of redshifts and mass scales (Lee et al., 2021).
Although Lee et al. (2022) showed that this peculiar phenomenon must be
closely linked with a physical process responsible for the deviation of the
galaxy stellar spin directions from their DM counterparts, its origin is still
shrouded in mystery, calling for more probing attentions and follow-up works.

The detection of this peculiar phenomenon has casted a doubt on the
conventional assumption that although the spin axes of the galaxy stellar
components are not perfectly in parallel with those of the DM counterparts
(e.g., Hahn et al., 2010; Velliscig et al., 2015; Tenneti et al., 2017; Zjupa
& Springel, 2017; Soussana et al., 2020), non-random orientations of the
former in the cosmic web must reflect the intrinsic alignments of the latter
(e.g., Codis et al., 2018, and references therein), the existence of which
have been theoretically predicted and numerically confirmed (for a
comprehensive review, see Joachimi et al., 2015). In fact, this conventional
assumption was adopted by many observational endeavors to find evidences for
the effect of the cosmic web on the evolution of the angular momentum of the
galactic halos by measuring the galaxy stellar spin orientations relative to
the cosmic web.

For example, Navarro et al. (2004) showed that the observed spin axes of the
spiral galaxies embedded in the Local Superclusters (LS) are inclined toward
the directions parallel to the LS plane, and suggested their observations as
an evidence supporting the linear tidal torque theory which generically
predicts the alignments between the spin axes of the galactic halos and the
intermediate principal axes of the tidal tensors (see also Trujillo et al.,
2006; Lee & Erdogdu, 2007). Jones et al. (2010) found a tendency of the spin
vectors of the edge-on spiral galaxies being aligned with the directions
perpendicular to the filaments in the local universe and interpreted this as a
”fossil evidence” of the existence of the large-scale tidal effect on the
angular momentum vectors of the galactic halos (see, also Kraljic et al.,
2021). Motloch et al. (2021) measured a significant signal of the alignments
between the spin axes of the local spirals and the principal axes of the
initial tidal field reconstructed from the spatial distributions of the local
galaxies from an all-sky survey, and declared it as a detection of the effect
of the initial tidal torques on the angular momentum of proto-galactic halos
(White, 1984; Lee & Pen, 2000, 2001).

However, if the observed non-random orientations of the galaxy stellar spins
with respect to the cosmic web do not take on the essential aspect of the
theoretically predicted intrinsic spin alignments of the galactic halos, as
hinted by Lee et al. (2022), then it requires a modification of the
conventional assumption and a new interpretation of the observational
evidences, which of course, should be preceded by physically understanding how
the galaxy stellar spins acquire the peculiar tidal connection. Among many
possible factors that may affect the evolution of the galaxy stellar spin
vectors and their deviations from the DM counterparts, here we consider the
merging events that most of the galaxies experience in their growths, in light
of the previous results based on hydrodynamical simulations that the alignment
strength and tendency of the galaxies depend strongly on their dynamics and
progenitor histories (Chisari et al., 2016; Bhowmick et al., 2020; Jagvaral et
al., 2022).

If the merging events make a substantial contribution to or at least have a
link with the generation of the peculiar spin alignments of the galaxy stellar
components, then the alignment strengths should sensitively depend on the
latest merger epochs. In this Paper, we are going to explore if and how the
peculiar stellar spin and shape alignments of the galaxies at $z=0$ vary with
their latest merger epochs with the help of a high-resolution hydrodynamical
simulation. Here, we focus only on the galaxies at the present epoch, $z=0$,
since the signal of the peculiar stellar spin alignments was found to be
significant only at low redshifts, $z<0.5$ (Lee et al., 2022). In addition, we
will also inspect the effects of merger events on the spin and shape
alignments of the galaxy DM, cold and hot gas components for comparison.

##  2 Numerical data and analysis

We utilize the data from the TNG300-1 simulation belonging to a series of
magnetohydrodynamical IllustrisTNG project (Marinacci et al., 2018; Naiman et
al., 2018; Nelson et al., 2018; Pillepich et al., 2018; Springel et al., 2018;
Nelson et al., 2019) conducted for a $\Lambda$CDM cosmology with the Planck
initial conditions (Planck Collaboration et al., 2016). Employing the
elaborate code, Arepo (Springel, 2010), the IllustrisTNG simulations were
capable of incorporating the full non-gravitational baryonic processes known
to play the key roles in driving the galaxy evolution. A detailed description
of the IllustrisTNG project can be found in its web page111https://www.tng-
project.org/data/.

The TNG300-1 simulation was performed in a periodic box of linear size
$302.6\,$Mpc containing $2500^{3}$ baryonic gas and equal number of DM
particles, the mass resolution of which are as high as $1.1\times
10^{7}\,M_{\odot}$ and $5.9\times 10^{7}\,M_{\odot}$, respectively. It found
the substructures of the friends-of-friends (FoF) groups via the Subfind
algorithm (Springel et al., 2001) and constructed their merger trees via the
SubLink code (Rodriguez-Gomez et al., 2015), the catalogs of which can be
extracted from the aforementioned webpage. Hereafter, we will refer to the
substructures of the FoF groups as the galaxies.

With the help of the routine provided by Lee et al. (2021), we first construct
the Gaussian-filtered tidal field, $T_{ij}({\bf x})$, on the $256^{3}$ grid
points at $z=0$. A brief summary of the routine is as follows: (i)
Constructing the mass density contrast field, $\delta({\bf x})$, on the
$256^{3}$ grid points by applying the cloud-in-cell method to the FoF halo
sample with no mass-cut at $z=0$. (ii) Performing the Fourier transformation
of $\delta({\bf x})$ to obtain its Fourier amplitude $\delta({\bf k})$ by
using the Fast Fourier Transformation (FFT) code (Press et al., 1992). (iii)
Computing the Fourier amplitude of the Gaussian-filtered tidal field as
$T_{ij}({\bf k})=\left[k_{i}k_{j}\delta({\bf
k})/k^{2}\right]\exp(-k^{2}R^{2}_{f}/2)$ where $k=|{\bf k}|$ and $R_{f}$ is
the filtering radius. (iv) Performing the inverse Fourier transformation of
$T_{ij}({\bf k})$ to obtain $T_{ij}({\bf x})$. As in Lee et al. (2022), we set
the filtering radius at $R_{f}=1.5\,$Mpc, the median size of cluster
environments in which the majority of the galaxies are embedded (e.g., Schmidt
et al., 1997; Tempel et al., 2014) . At the grid that matches the position of
each galaxy, we carry out the similarity transformation of $T_{ij}({\bf x})$
to find its major, intermediate and minor principal axes (${\bf e}_{1},\ {\bf
e}_{2}$ and ${\bf e}_{3}$, respectively), corresponding to the largest, second
largest and smallest eigenvalues ($\lambda_{1},\ \lambda_{2}$ and
$\lambda_{3}$, respectively). The major (minor) principal axes of the local
tidal tensors are parallel to the directions of maximum (minimum) matter
compression.

For the investigation of the merger effects on the spin and shape alignments
of the galaxies with the principal directions of the local tidal fields, we
separately consider the four components of each galaxy, namely, stellar, cold
gas, hot gas and DM. Here, the cold (hot) gas component corresponds to the
non-stellar baryons with temperature, $T$, lower (higher) than $10^{5}$ K,
above which the gas is mostly created by gravitational shock heating or
feedback processes (e.g., Martizzi et al., 2019). We determine the temperature
of each non-stellar gas cell by converting its specific internal energy and
electron abundance, information on which are all available in the TNG300-1
snapshot data. The public data release from the IllustrisTNG project also
provides all other necessary information for our investigation, such as the
comoving positions and velocities of all member particles, numbers of the cold
gas, hot gas and DM components of each galaxy ($n_{cg}$, $n_{hg}$, and
$n_{d}$, respectively), and total mass of all member particles ($M_{\rm
tot}$), and numbers of the stellar cells ($n_{s}$) within twice its stellar
half-mass radius ($2R_{1/2}$) at $z=0$.

For each of the galaxies with $n_{s}\geq 300$, we compute the spin vector of
its stellar component, ${\bf J}_{s}$, as

${\bf J}_{s}=\sum_{\alpha=1}^{n_{s}}{m}_{s,\alpha}\,[({\bf x}_{s,\alpha}-{\bf
x}_{c})\times({\bf v}_{s,\alpha}-{\bf v}_{c})]\,,\\\ $ (1)

where $m_{s,i}$, ${\bf x}_{s,\alpha}=(x_{s,\alpha\,i})$ and ${\bf
v}_{s,\alpha}$ are the mass, comoving position and peculiar velocity of the
$\alpha$th stellar particle within $2R_{1/2}$, while ${\bf x}_{c}=(x_{c,i})$
and ${\bf v}_{c}$ are the positions and velocities of the galaxy center,
respectively. The number cut of the stellar particles, $n_{s}\geq 300$, is
applied to the galaxies, since the measurements of the stellar spin vectors
from the lower number of particles would suffer from low accuracy (Bett et
al., 2007). The unit stellar spin vector, ${\bf j}_{s}$, is then calculated as
${\bf j}_{s}\equiv{\bf J}/|{\bf J}|$.

The stellar inertia tensor, ${\bf I}_{s}=(I_{s,ij})$, of each selected galaxy
with $n_{s}\geq 300$ is computed as (Cole & Lacey, 1996):

$I_{s,ij}=\frac{1}{M_{s}}\sum_{\alpha=1}^{n_{s}}{m}_{s,\alpha}\,(x_{s,\alpha\,i}-x_{c,i})\,(x_{s,\alpha\,j}-x_{c,j})\,,\quad
i,j\in\\{1,\ 2,\ 3\\}\,,$ (2)

where $M_{s}\equiv\sum_{\alpha=1}^{n_{s}}{m}_{s,\alpha}$. The similarity
transformation of ${\bf I}_{s}$ yields a set of orthonormal eigenvectors
corresponding to its three eigenvalues (say, $q_{1},q_{2},q_{3}$ in a
decreasing order). We define the unit stellar shape vector of a galaxy, ${\bf
u}_{s}$, as the eigenvector of ${\bf I}_{s}$ corresponding to $q_{3}$, which
is in parallel to the direction of the minor axis of the galaxy stellar shape.
We also quantify the morphological shape of the stellar distribution of each
selected galaxy by measuring its degrees of triaxiality (${\cal T}_{s}$) and
sphericity (${\cal S}_{s}$) (Franx et al., 1991; Bett et al., 2007):

${\cal T}_{s}=\frac{q_{1}-q_{2}}{q_{1}-q_{3}},\qquad{\cal
S}_{s}=\left(\frac{q_{3}}{q_{1}}\right)^{1/2}\,.$ (3)

The oblate (prolate) shape corresponds to ${\cal T}_{s}<0.5$ (${\cal
T}_{s}>0.5$), while the more aspherical shape has a lower value of ${\cal
S}_{s}$. Separately applying three different particle number cuts, $n_{cg}\geq
300$, $n_{hg}\geq 300$, $n_{d}\geq 300$, we determine the unit spin and shape
vectors, $\\{{\bf j}_{cg},{\bf u}_{cg}\\}$, $\\{{\bf j}_{hg},{\bf u}_{hg}\\}$,
$\\{{\bf j}_{d},{\bf u}_{d}\\}$, and morphological shapes, $\\{{\cal
T}_{cg},{\cal S}_{cg}\\}$, $\\{{\cal T}_{hg},{\cal S}_{hg}\\}$, $\\{{\cal
T}_{d},{\cal S}_{d}\\}$, of each selected galaxy in a similar manner.

At this stage, it is worth explaining here why we include only those stellar
components within $2R_{1/2}$, rather than using all of them contained in the
host subhalos for the measurements of ${\bf j}_{s}$ and $I_{s,ij}$. The main
reason is to make our numerical analysis in line with the observational ones.
For the determination of the spin orientations of the real galaxies from
observations, it is necessary to measure their kinetic properties. But, it has
been known for long that measuring them outside $2R_{1/2}$ is very difficult
in practice (Romanowsky et al., 2003; Coccato et al., 2009; Welker et al.,
2020). For this very reason, most of the spectroscopic galaxy surveys like the
Institute For Astronomy Survey (IFA), the Mapping Nearby Galaxies at APO
survey (MaNGA), the Sydney-AAO Multi-object Integral Field Spectrograph Galaxy
Survey (SAMI), and the Calar Alto Legacy Integral Field Area Survey (CALIFA)
were originally designed to observe the stellar components within $2R_{1/2}$
(Krolewski et al., 2019; Lee et al., 2019; Welker et al., 2020). Moreover,
given that the spin parameters, $\lambda$, of the real galaxies from the
observations are usually measured from the stellar components within
$2R_{1/2}$ (Emsellem et al., 2007; Cortese et al., 2016), it should be quite
reasonable and appropriate to adopt the same radial distance cutoff,
$2R_{1/2}$, for the determination of the stellar spin directions 222In real
observations, $R_{1/2}$ is defined as the half luminosity radius rather than
the half stellar mass radius (Romanowsky et al., 2003)..

Tracking back the main progenitor branch of the merger tree for each galaxy in
the TNG300-1, we determine the scale factor of the epoch, say $a(z_{m})$, when
its latest merger event occurs. For this analysis, we consider only those
mergers events in which the involved galaxies have stellar mass ratios larger
than $0.1$ (i.e., both major and minor mergers) and exclude those galaxies
which never experience any merger event during the evolution. The merger mass
ratio is calculated from the past maximum stellar masses of each merging
galaxy to avoid the notorious artificial mass loss problem in the halo finding
algorithm (Rodriguez-Gomez et al., 2015). Splitting the galaxies into four
equal-size samples according to the values of $a(z_{m})$, we control the four
samples to have identical joint distributions of $\log\tilde{M}_{\rm tot}$ and
$\log\,(1+\delta)$, where $\tilde{M}_{\rm tot}\equiv M_{\rm
tot}/(10^{7}\,h^{-1}\,M_{\odot})$ and $\delta\equiv\sum_{i=1}^{3}\lambda_{i}$,
for the purpose of nullifying the possible effects of the differences in the
total masses of the galaxies and the environmental densities among the four
samples on the alignment strengths, as done in Lee et al. (2022).

We calculate the following alignments ensemble averaged over each of the
controlled samples (say, $A$, $B$, $C$ and $D$), and investigate the
variations of their strengths with $a(z_{m})$: the spin alignments between the
baryon and DM components ($\langle{\bf j}_{s}\cdot{\bf j}_{d}\rangle$,
$\langle{\bf j}_{cg}\cdot{\bf j}_{d}\rangle$, $\langle{\bf j}_{hg}\cdot{\bf
j}_{d}\rangle$); the alignments between the baryon spins and major principal
axes of the local tidal tensors called the spin-shear alignments
($\langle|{\bf j}_{s}\cdot{\bf e}_{1}|\rangle$, $\langle|{\bf j}_{cg}\cdot{\bf
e}_{1}|\rangle$, $\langle|{\bf j}_{hg}\cdot{\bf e}_{1}|\rangle$); and the
alignments between the DM spins of high-mass (low-mass) galaxies and
intermediate (minor) principal axes of the local tidal tensors ($\langle|{\bf
j}_{d}\cdot{\bf e}_{1}|\rangle$, $\langle|{\bf j}_{d}\cdot{\bf
e}_{2}|\rangle$, $\langle|{\bf j}_{d}\cdot{\bf e}_{3}|\rangle$); the
alignments between the spin and shape vectors of each component called the
spin-shape alignments ($\langle|{\bf j}_{s}\cdot{\bf u}_{s}|\rangle$,
$\langle|{\bf j}_{cg}\cdot{\bf u}_{cg}|\rangle$, $\langle|{\bf
j}_{hg}\cdot{\bf u}_{hg}|\rangle$, $\langle|{\bf j}_{d}\cdot{\bf
u}_{d}|\rangle$); the alignments between the shape vectors and tidal major
principal directions called the shape-shear alignments ($\langle|{\bf
u}_{s}\cdot{\bf e}_{1}|\rangle$, $\langle|{\bf u}_{cg}\cdot{\bf
e}_{1}|\rangle$, $\langle|{\bf u}_{hg}\cdot{\bf e}_{1}|\rangle$, $\langle|{\bf
u}_{d}\cdot{\bf e}_{1}|\rangle$). We also compute the standard deviation in
each ensemble average as the associated error. For instance, the error in
$\langle{\bf j}_{s}\cdot{\bf j}_{d}\rangle$ is computed as
$\sqrt{\left[\langle({\bf j}_{s}\cdot{\bf j}_{d})^{2}\rangle-\langle{\bf
j}_{s}\cdot{\bf j}_{d}\rangle^{2}\right]/(N_{g}-1)}$ where $N_{g}$ is the
number of the galaxies belonging to each sample.

Note that the spin alignments between the baryon and DM components are
measured as the dot-products between the spin vectors while the spin-shear
alignments are measured as the absolute values of the dot-products between the
galaxy spin vectors and the principal axes of the local tidal tensors. It is
simply because for the latter case there is no difference in the physical
meaning between the positive and negative signs of the dot-product, while for
the former case there is. For instance, suppose that there is a galaxy whose
stellar components are spinning clockwise when viewed face on. If ${\bf
j}_{s}\cdot{\bf j}_{d}=-1$, then it indicates counter-clockwise spinning of
the galaxy DM components. It is also worth mentioning here that the criteria
on $a_{m}(z)$ used to separate the four samples depend on which quantity to
calculate, since different numbers of galaxies are selected by different
particle number cuts to calculate the alignments of different components. For
example, for the calculation of $\langle{\bf j}_{s}\cdot{\bf j}_{d}\rangle$,
only those galaxies that satisfy both of the criteria of $n_{s}\geq 300$ and
$n_{d}\geq 300$ are selected and divided into the four samples. For the
calculation of $\langle|{\bf j}_{s}\cdot{\bf e}_{1}|\rangle$, the single
condition of $n_{s}\geq 300$ is used to separate the samples.

##  3 Dependence on the latest merger epochs

###  3.1 Baryon spin-DM spin alignments

Figure 1 shows how the mean strengths of the alignments of the stellar, cold
and hot gas spin axes of the galaxies with their DM counterparts depend on the
latest merger epochs, $a(z_{m})$, at $z=0$ (top, middle and bottom panels,
respectively). As can be seen, the earlier the latest merger events occur, the
lower the mean alignments between the galaxy baryon and DM spin vectors
become. Note that for the case of the galaxies with $a(z_{m})<0.4$, the value
of $\langle{\bf j}_{s}\cdot{\bf j}_{d}\rangle$ drops below $0.5$, the
expectation value from two randomly oriented unit vectors. From here on, we
refer to this tendency of having the ensemble average of the dot-product
between two unit vectors below $0.5$ as anti-alignment. The result shown in
the top panel of Figure 1 implies that the mechanism responsible for this
${\bf j}_{s}$-${\bf j}_{d}$ anti-alignment must occur only in the quiescent
period of time when no merging event occurs.

Unlike the stellar components, both of the cold and hot gas components exhibit
no anti-alignment but only alignment tendency with the DM counterparts
($\langle{\bf j}_{cg}\cdot{\bf j}_{d}\rangle>0.5$ and $\langle{\bf
j}_{hg}\cdot{\bf j}_{d}\rangle>0.5$). The more recently the latest merger
events occur, the stronger the gas-DM spin alignments become, which implies
that the merger events have an effect of aligning the baryon spin axes of the
galaxies toward the DM counterparts. Noting also that the ${\bf j}_{cg}$-${\bf
j}_{d}$ and ${\bf j}_{hg}$-${\bf j}_{d}$ alignments exhibit little difference
in strength between each other, we suspect that the cooling process in the
cold gas should have little effect of weakening the gas-DM spin alignments.
Figure 2 and 3 show how well the four $a(z_{m})$-selected samples are
controlled to yield almost identical mean values of $\log\tilde{M}$ and
$\log(1+\delta)$ for three differences cases of the particle number cuts. It
is now guaranteed that the different strengths of the baryon-DM spin
alignments among the four $a(z_{m})$-selected samples, as witnessed in Figure
1, are caused not by any difference in the galaxy total masses and
environmental densities.

###  3.2 Spin-shear alignments

Figure 4 shows how the spin orientations of the galaxy stellar, cold and hot
gas components relative to the major principal directions of the local tidal
tensors differ among the four $a(z_{m})$-selected samples (top, middle and
bottom panels, respectively). A significant alignment signal is found only
from the stellar spin axes of the galaxies whose latest merger events occur at
$a(z_{m})<0.4$. The earlier the latest merger events occur, the stronger the
${\bf j}_{s}$-${\bf e}_{1}$ alignments become, provided that $a(z_{m})<0.4$.
For the case of the galaxies that experience more recent merger events at
$a(z_{m})\geq 0.4$, however, their stellar spin axes seem to be randomly
oriented with respect to the major principal directions of the tidal field.

This result implies that the ${\bf j}_{s}$-${\bf e}_{1}$ alignments should be
generated by some physical process during the quiescent evolution period, and
that the occurrence of the merger events plays the role of destroying the
${\bf j}_{s}$-${\bf e}_{1}$ alignments. A comparison with the results shown in
Figure 1 leads us to think that the ${\bf j}_{s}$-${\bf j}_{d}$ anti-
alignments and the ${\bf j}_{s}$-${\bf e}_{1}$ alignments must be generated by
the same physical process, as envisaged by Lee et al. (2022). In contrast, the
non-stellar gas spin axes of the galaxies yield only anti-alignment tendencies
with the major principal directions of the tidal field, regardless of
$a(z_{m})$. For the case of the cold gas, the anti-alignment strength
increases with $a(z_{m})$, while for the case of the hot gas, no significant
variation with $a(z_{m})$ is found.

The top panel of Figure 5 shows how the DM spin orientations relative to the
major principal directions of the tidal tensors differ among the four
$a(z_{m})$-selected samples (top, middle and bottom panels, respectively). As
can be seen in the top panel, the ${\bf j}_{d}$-${\bf e}_{1}$ anti-alignment
does not exhibit a strong variation with the latest merger epochs for the case
of $a(z_{m})<0.66$, but a substantially weaker signal is found for the case of
$a(z_{m})\geq 0.66$, similar to the ${\bf j}_{hg}$-${\bf e}_{1}$ anti-
alignment (bottom panel of Figure 4). A comparison of this result with those
shown in Figure 4 casts two crucial implications. First, the merger events
have only momentary weak effect of undermining the ${\bf j}_{hg}$-${\bf
e}_{1}$ and ${\bf j}_{d}$-${\bf e}_{1}$ anti-alignment tendencies. Second, the
hot gas and DM spin orientations should be insensitive to the physical process
responsible for the generation of the ${\bf j}_{s}$-${\bf e}_{1}$ alignments
during the quiescent evolution period, while the merger events have an
indirect effect of weakening the ${\bf j}_{s}$-${\bf e}_{1}$ alignments and
${\bf j}_{cg}$-${\bf e}_{1}$ anti-alignments by ending the quiescent period.

Recall that the DM spin axes exhibit conspicuous mass-dependent transition
phenomena: In the high-mass section, the DM spin orientations are
preferentially aligned with the intermediate principal axes of the tidal
fields while in the low-mass section they tend to be aligned with the minor
principal directions (Lee et al., 2021). Taking into account the occurrence of
the DM spin transition phenomena, we divide the galaxies into the high-mass
and low-mass groups containing those galaxies with $n_{d}\geq 10^{4}$ and
$300\leq n_{d}<10^{4}$, respectively. Then, we investigate separately how the
DM spin orientations of the high-mass and low-mass galaxies relative to the
intermediate and minor principal axes vary with the latest merger epochs, the
results of which are plotted in the middle and bottom panels of Figure 5,
respectively. As can be seen, the strengths of the ${\bf j}_{d}$-${\bf e}_{2}$
and ${\bf j}_{d}$-${\bf e}_{3}$ alignments do not show any significant
variation with the latest merger epochs, which implies that the merger events
are not responsible for the mass-dependent transition of the DM spin vectors
relative to the principal directions of the tidal field. Figures 6-7 plots the
same as Figures 2-3 but for four different cases of the particle number cuts,
proving that the four controlled samples have no differences in the total mass
and environmental density distributions, and thus that the detected trend of
the spin-shear alignments is truly caused by the differences in the latest
merger epochs.

###  3.3 Spin-shape and shape-shear alignments

Figure 8 shows how the strengths of the spin-shape alignments of the galaxy
stellar, cold and hot gas, and DM components depend on the latest merger
epochs (top, second from the top, second from the bottom, bottom panels,
respectively). As can be seen in the top panel, the stellar components yield
the strongest spin-shape alignments whose strength monotonically decreases
with $a(z_{m})$, The earlier the latest merger events occur, the stronger the
${\bf j}_{s}$-${\bf u}_{s}$ alignments become. Note that the galaxies whose
latest merger epochs occur at $a(z_{m})\leq 0.25$ yield $\langle|{\bf
j}_{s}\cdot{\bf u}_{s}|\rangle\sim 1$, signaling the existence of almost
perfect stellar spin-shape alignments. The physical process responsible for
the peculiar ${\bf j}_{s}$-${\bf e}_{1}$ alignments in the quiescent period
must generate this almost perfect stellar spin-shear alignments, while the
mergers ruin this strong alignments by interrupting the quiescent period.

As shown in the bottom panel of Figure 8, the variation of the DM spin-shape
alignment with $a(z_{m})$ exhibits a directly opposite trend, increasing as
the latest merger events occur more recently. In other words, the mergers
drive the DM components to have shapes whose minor axes become more strongly
aligned with their spin vectors. A comparison with the result shown in the top
panel of Figure 8 leads us to think that the spin-shape alignments can be
enhanced by two different mechanisms, the mergers and the physical process
responsible for the ${\bf j}_{s}$-${\bf e}_{1}$ alignments. The DM components
are insensitive to the latter mechanism, while the stellar components are
sensitive to both of the mechanisms. Although the former mechanism has a
direct effect of enhancing the spin-shape alignment even of stellar component,
it also produces an indirect opposite effect by ending the quiescent period,
only during which the latter mechanism effectively enhances the stellar spin-
shape alignments. This scenario explains why the stellar spin-shape alignments
decrease with $a(z_{m})$, while the DM counterparts exhibit an opposite trend.

Meanwhile, the spin-shape alignments of the cold and hot gas components yield
non-monotonic variations with the latest merger epochs, as shown in the second
and third panels from the bottom of Figure 8. This result indicates that the
non-stellar gas components are affected by both of the mechanisms and that the
ranges of $a(z_{m})$ at which the weakest spin-shape alignments are found
correspond to the epochs when the two mechanisms counter-balance each other.
Note that the hot gas components seem to be more vulnerable to the effect of
the mergers while on the cold gas counterparts is more dominant the effect of
the physical process responsible for the ${\bf j}_{s}$-${\bf e}_{1}$
alignments.

Figure 9 shows how the shape-shear alignments of the stellar, cold and hot
gas, and DM components depend on the latest merger epochs (top, second from
the top, second from the bottom, bottom panels, respectively). As can be seen,
the shape vectors of the gas and DM components tend to align with the major
principal axes of the tidal tensors to which their spin vectors tend to be
perpendicular. This result implies that the shape axes of the gas and DM
components are not good tracers of their spin directions. The shape-shear
alignments of all of the four components decrease almost monotonically with
$a(z_{m})$, which indicates that the mergers have an effect of deviating the
galaxy shape directions from the major principal axes of the local tidal
tensors, regardless of the components. It is the DM components that exhibit
the strongest shape-shear alignments while the weakest alignments are found
from the cold gas components.

Witnessing that the signals of the shape-shear alignments are significant in
the entire range of $a(z_{m})$ for the cases of the hot gas and DM, while the
cold gas and stellar components yield no significant signals at
$a(z_{m})>0.65$, we suspect that the cooling process of baryon particles occur
most efficiently along the directions of maximum matter compression, which in
consequence deviate the shape vectors of the cold gas and stellar components
(i.e., the minor axes of the inertia momentum tensors) from the major
principal axes of the local tidal tensors. Note also that the strengths and
$a(z_{m})$-dependence of the shape-shear alignments of the galaxy stellar
components are very similar to those of their spin-shear alignments, as
naturally expected from our finding of the very strong ${\bf j}_{s}$-${\bf
u}_{s}$ alignments (see Figure 8).

###  3.4 Triaxiality and sphericity

Figures 10 and 11 show how the merger events affect the morphological shapes
of the galaxy stellar, cold and hot gas, and DM components (top, second from
the top, second from the bottom, and bottom panels, respectively). As can be
seen, the galaxy stellar components have conspicuously oblate shapes (${\cal
T}_{s}<0.5$), unlike the other three components all of which exhibit prolate
shapes (${\cal T}_{cg},\ {\cal T}_{hg},\ {\cal T}_{d}>0.5$). The earlier the
latest merger events occur, the more oblate shapes the galaxy stellar
components have, indicating that the galaxy stellar components develop oblate
shapes during the quiescent period, but evolve into less oblate and more
elongated shapes through the mergers. The mergers also reorganize the galaxy
cold gas and DM distributions of the galaxies to have more prolate and
aspherical shapes, while the hot gas component of the galaxies seem to be
least susceptible to the merger events, with the mean values of ${\cal
T}_{hg}$ and ${\cal S}_{hg}$ varying only weakly with $a(z_{m})$.

Among the four components, it is the cold gas components that have the most
prolate and most aspherical shapes, while it is the stellar components that
have the most oblate shapes. Note also that the cold gas components are more
aspherical than the stellar counterparts, with ${\cal S}_{cg}$ decreasing
almost monotonically with $a(z_{m})$. Given this palpable difference between
the morphologies of the stellar and cold gas components, we suspect that the
cooling process in the cold gas should not directly contribute to the
generations of the peculiar ${\bf j}_{s}$-${\bf e}_{1}$ alignment and the
strongest ${\bf j}_{s}$-${\bf u}_{s}$ alignments. In other words, the physical
process through which the gas particles cool down and evolve into stars is
unlikely to be the main mechanism that drives the stellar spin and shape
vectors to align with the directions of maximum matter compression. If the gas
cooling process were to be the main mechanism, the morphologies of the cold
gas components would be similar to those of the stellar counterparts, having
less prolate shapes than the hot gas and DM components.

##  4 Summary and discussion

Analyzing separately the stellar, cold and hot gas, and DM components of the
well-resolved galaxies from the TNG 300-1 simulation of the IllustrisTNG
project, we have explored how the strengths and tendencies of the spin-shear,
spin-shape, and shape-shear alignments of each component depend on the latest
merger epochs at $z=0$. The effects of the environmental density and total
mass differences on the strengths of these alignments have been properly
eliminated by controlling the galaxy samples. The key results from this
exploration and their implications are summarized in the following.

  * •

The peculiar alignments of the galaxy stellar spins with the major principal
axes of the tidal field and their anti-alignments with the DM counterparts can
be developed only during the quiescent evolution period when the galaxies
experience no recent mergers (Figures 1-4). The merger events have an effect
of destroying the peculiar stellar spin alignments by orienting the stellar
spin axes toward the DM counterparts that are preferentially aligned with the
directions perpendicular to the major principal axes of the local tidal
tensors. The physical process responsible for the generation of this peculiar
stellar spin alignments must affect only the stellar and cold gas components
but not the hot gas and DM components.

  * •

No strong variation of the strengths with the latest merger epoch is exhibited
by the DM spin alignments of high-mass and low-mass galaxies with the
intermediate and minor principal directions of the local tidal tensors,
respectively (Figures 5-7), which implies that the merger events are not the
primary mechanism for the generation of the spin transition phenomenon of
galactic halos (c.f., Pichon et al., 2011; Codis et al., 2012; Dubois et al.,
2014; Krolewski et al., 2019).

  * •

During the quiescent period, the galaxy stellar components develop oblate
shapes whose minor axes are almost perfectly aligned with their spin
directions (Figures 8-10). The stellar shape-shear alignments mirror very well
the stellar spin-shear alignments in strengths and behaviors as a function of
the latest merger epoch (Figure 9). The physical process responsible for the
peculiar stellar spin alignments should be closely linked with the almost
perfect stellar spin-shape alignments.

  * •

The non-stellar gas and DM components have prolate shapes whose minor axes are
not so perfectly aligned with their spin directions as the stellar components.
The merger events enhance the DM spin-shape alignments, while weakening the
stellar counterparts (Figure 8). This striking difference between the stellar
and DM components is likely caused by the dual roles of a merger event.
Although it has the effect of enhancing the spin-shape alignments of each
component, the occurrence of a merger event ends the quiescent period at the
same time, which incapacitates the same enhancing effect of the physical
process on the stellar spin-shape alignments. The duality of the merger
effects is well reflected by the non-monotonic variation of the cold gas spin-
shape alignments with the latest merger epochs.

Our results provides a counter evidence against the prevalent scenario that
the spin transition of galactic halos should be ascribed to the merger effects
(Aragón-Calvo et al., 2007; Hahn et al., 2007; Pichon et al., 2011; Codis et
al., 2012; Trowland et al., 2013; Dubois et al., 2014; Codis et al., 2018;
Ganeshaiah Veena et al., 2018; Krolewski et al., 2019). According to this
scenario, the spin axes of high-mass galaxies acquire a tendency of being
aligned with the directions perpendicular to the host filaments, as they
evolve through anisotropic mergers along the filaments. Whereas the low-mass
galaxies have their spin directions aligned with the directions parallel to
the filaments, as they undergo not so frequent mergers as the high-mass
counterparts. If this scenario were true, then the strengths of the spin
alignments of the high-mass galaxies with the intermediate principal
directions of the tidal tensors (perpendicular to the filament axes) would
yield a signal of significant variation with the latest merger epochs.
However, no such signal has been detected by our analysis, which indicates
that the mergers contribute very little to establishing the spin transition of
the galactic halos.

Rather, our finding is consistent with the tidal torque picture that the DM
spin alignments of high-mass galaxies with the filament axes originate from
the tidal interactions at the proto-galactic stages (Lee & Pen, 2000, 2001;
Jones et al., 2010; Motloch et al., 2021; Lee et al., 2021, 2022). In the
subsequent evolution, the occurrence of mergers could significantly change the
DM spin directions of galactic halos without destroying the tidally induced
alignments, since the DM spin angular momenta of the merged halos are in fact
the transferred orbital angular momenta that are also aligned with the
intermediate principal axes of the tidal field on larger scales. In other
words, the merger events change the scale of the tidal field whose principal
axes the DM spin directions are aligned with. Regarding the DM spin alignments
of low-mass galaxies with the minor principal axes, the tidal torque theory
provides no physical answer to the critical question of what mechanism
produces it. However, given our result that their strengths also exhibit no
variation with the latest merger epochs, their origin should not be the
occurrence of the merger events, either, as suspected by Lee et al. (2020).

Our result also challenges the conventional assumption that the minor axes of
the galaxy shapes is a good proxy of their spin directions. Under this
assumption, the observed stellar shape alignments of the galaxies with the
cosmic web were often regarded and interpreted in the previous works as the
evidences for the existence of the DM spin alignments of galactic halos (e.g.,
Jones et al., 2010; Lee et al., 2018; Krolewski et al., 2019; Lee, 2019;
Welker et al., 2020; Kraljic et al., 2021; Motloch et al., 2021), since the
former is more readily observable while the latter is what a theory or a
numerical experiment can model and predict. As revealed by our work, however,
only for the case of the galaxy stellar components, their shape-shear
alignments mirror well the spin-shear counterparts, while for the case of the
non-stellar gas and DM components of the galaxies, the former notably differs
from the latter, especially in variation with the latest merger epochs. In
other words, the theoretical model for the DM spin-shear alignments of
galactic halos should not be directly compared with the observed stellar
shape-shear alignments, since the two phenomena are likely to originate from
different mechanisms.

Yet, it is worth mentioning that our results on the peculiar tidal connections
are contingent upon our choice of $2R_{1/2}$ as the radial distance cut-off
for the determinations of the galaxy stellar spin directions. As previously
shown in Chisari et al. (2016), the strength and tendency of the stellar spin
and shape alignments of the galaxies sensitively vary with the choice of the
cut-off or weights on the radial distances of the stellar components (see,
also Jagvaral et al., 2022). The discrepancy of our result with the previous
works which found quite similar tendency between the stellar and DM spin axes
of the galaxies in their alignments with the cosmic web (Codis et al., 2018)
may be caused by the difference in the way that the stellar angular momentum
vectors were measured. Unlike the previous work where the subhalos having $50$
or more stellar components without putting any radial distance cutoff, the
current analysis has taken a more conservative approach in line with the
observational analyses, including only those subhalos having $300$ or more
stellar components within $2R_{1/2}$.

Nevertheless, the existence of the peculiar tidal connection of the stellar
spins of the galaxies in quiescent evolution and their almost perfect spin-
shape alignments hints that it might be even more plausible to reconstruct the
local tidal and density fields from the observable stellar shapes of the
galaxies with an algorithm similar to the one developed by Lee & Pen (2000)
and recently used in practice by Motloch et al. (2021). In these previous
works, the weak alignments between the DM spin axes of the galaxies and the
intermediate principal axes of the tidal fields is assumed to reflect well the
anisotropic tidal fields. This algorithm, however, has a limited usefulness
due to several practical difficulties: First, the DM spin axes are not readily
observable unlike the stellar counterparts. Second, the alignments between the
DM spins and the intermediate principal axes of the tidal fields can be found
only from the high-mass galaxies but not from the low-mass galaxies due to the
spin transition phenomena. Third, in the sheet and filamentary environments,
the intermediate principal axes of the tidal fields are degenerated with the
other principal axes.

If this algorithm could be modified to accommodate the peculiar stellar spin
alignments of the galaxies with the major principal axes of the tidal fields
and to use it for the reconstruction of the tidal and density fields, then the
aforementioned practical difficulties would be avoided. The stellar shape
alignments of the galaxies are readily observable, the peculiar stellar spin
alignments do not exhibit transition phenomena on the galactic scale, and the
major principal axes of the tidal fields do not suffer from the degeneracy
even in the filamentary or sheet environments. Our future work will be in the
direction of developing such an algorithm.

In conclusion, we have found several key clues to the physical process that
generates the peculiar alignments of the galaxy stellar spins with the major
principal axes of the local tidal field. We believe that these clues will
provide deeper insights to understand and model the physical process, which
must precede any attempt to connect the observable shape-shear alignments of
the galaxy stellar components to the initial conditions of the universe (e.g.,
Lee & Pen, 2000; Taruya & Okumura, 2020; Motloch et al., 2021).

We thank an anonymous referee whose thoughtful review helped us improve the
original manuscript. The IllustrisTNG simulations were undertaken with compute
time awarded by the Gauss Centre for Supercomputing (GCS) under GCS Large-
Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of the supercomputer
Hazel Hen at the High Performance Computing Center Stuttgart (HLRS), as well
as on the machines of the Max Planck Computing and Data Facility (MPCDF) in
Garching, Germany. JL acknowledges the support by Basic Science Research
Program through the National Research Foundation (NRF) of Korea funded by the
Ministry of Education (No.2019R1A2C1083855).

## References

  * Aragón-Calvo et al. (2007) Aragón-Calvo, M. A., van de Weygaert, R., Jones, B. J. T., et al. 2007, ApJ, 655, L5
  * Aragon-Calvo & Yang (2014) Aragon-Calvo, M. A., & Yang, L. F. 2014, MNRAS, 440, L46
  * Bett et al. (2007) Bett, P., Eke, V., Frenk, C. S., et al. 2007, MNRAS, 376, 21
  * Bhowmick et al. (2020) Bhowmick, A. K., Chen, Y., Tenneti, A., et al. 2020, MNRAS, 491, 4116
  * Chisari et al. (2016) Chisari, N., Laigle, C., Codis, S., et al. 2016, MNRAS, 461, 2702
  * Coccato et al. (2009) Coccato, L., Gerhard, O., Arnaboldi, M., et al. 2009, MNRAS, 394, 1249
  * Codis et al. (2012) Codis, S., Pichon, C., Devriendt, J., et al. 2012, MNRAS, 427, 3320
  * Cole & Lacey (1996) Cole, S. & Lacey, C. 1996, MNRAS, 281, 716
  * Codis et al. (2018) Codis, S., Jindal, A., Chisari, N. E., et al. 2018, MNRAS, 481, 4753
  * Cortese et al. (2016) Cortese, L., Fogarty, L. M. R., Bekki, K., et al. 2016, MNRAS, 463, 170
  * Deason et al. (2012) Deason, A. J., Belokurov, V., Evans, N. W., et al. 2012, ApJ, 748, 2
  * Dubois et al. (2014) Dubois, Y., Pichon, C., Welker, C., et al. 2014, MNRAS, 444, 1453
  * Emsellem et al. (2007) Emsellem, E., Cappellari, M., Krajnović, D., et al. 2007, MNRAS, 379, 401
  * Franx et al. (1991) Franx, M., Illingworth, G., & de Zeeuw, T. 1991, ApJ, 383, 112
  * Hahn et al. (2007) Hahn, O., Carollo, C. M., Porciani, C., et al. 2007, MNRAS, 381, 41
  * Hahn et al. (2010) Hahn, O., Teyssier, R., & Carollo, C. M. 2010, MNRAS, 405, 274
  * Ganeshaiah Veena et al. (2018) Ganeshaiah Veena, P., Cautun, M., van de Weygaert, R., et al. 2018, MNRAS, 481, 414
  * Ganeshaiah Veena et al. (2019) Ganeshaiah Veena, P., Cautun, M., Tempel, E., et al. 2019, MNRAS, 487, 1607
  * Jagvaral et al. (2022) Jagvaral, Y., Singh, S., & Mandelbaum, R. 2022, MNRAS, 514, 1021
  * Joachimi et al. (2015) Joachimi, B., Cacciato, M., Kitching, T. D., et al. 2015, Space Sci. Rev., 193, 1
  * Jones et al. (2010) Jones, B. J. T., van de Weygaert, R., & Aragón-Calvo, M. A. 2010, MNRAS, 408, 897
  * Kraljic et al. (2020) Kraljic, K., Davé, R., & Pichon, C. 2020, MNRAS, 493, 362
  * Kraljic et al. (2021) Kraljic, K., Duckworth, C., Tojeiro, R., et al. 2021, MNRAS, 504, 4626
  * Kristensen et al. (2021) Kristensen, M. T., Pimbblet, K. A., Gibson, B. K., et al. 2021, ApJ, 922, 127
  * Krolewski et al. (2019) Krolewski, A., Ho, S., Chen, Y.-C., et al. 2019, ApJ, 876, 52
  * Lee & Pen (2000) Lee, J., & Pen, U.-L. 2000, ApJ, 532, L5
  * Lee & Pen (2001) Lee, J., & Pen, U.-L. 2001, ApJ, 555, 106
  * Lee & Erdogdu (2007) Lee, J. & Erdogdu, P. 2007, ApJ, 671, 1248
  * Lee et al. (2018) Lee, J., Kim, S., & Rey, S.-C. 2018, ApJ, 860, 127
  * Lee (2019) Lee, J. 2019, ApJ, 872, 37
  * Lee et al. (2019) Lee, J. H., Pak, M., Song, H., et al. 2019, ApJ, 884, 104
  * Lee et al. (2020) Lee, J., Libeskind, N. I., & Ryu, S. 2020, ApJ, 898, L27
  * Lee & Libeskind (2020) Lee, J. & Libeskind, N. I. 2020, ApJ, 902, 22
  * Lee et al. (2021) Lee, J., Moon, J.-S., Ryu, S., et al. 2021, ApJ, 922, 6
  * Lee et al. (2022) Lee, J., Moon, J.-S., & Yoon, S.-J. 2022, ApJ, 927, 29
  * Libeskind et al. (2013) Libeskind, N. I., Hoffman, Y., Forero-Romero, J., et al. 2013, MNRAS, 428, 2489
  * Marinacci et al. (2018) Marinacci, F., Vogelsberger, M., Pakmor, R., et al. 2018, MNRAS, 480, 5113
  * Martizzi et al. (2019) Martizzi, D., Vogelsberger, M., Artale, M. C., et al. 2019, MNRAS, 486, 3766
  * Moon et al. (2021) Moon, J. S., An, S. H., & Yoon, S.-J. 2021, ApJ, 909, 34
  * Motloch et al. (2021) Motloch, P., Yu, H.-R., Pen, U.-L., et al. 2021, Nature Astronomy, 5, 283
  * Naiman et al. (2018) Naiman, J. P., Pillepich, A., Springel, V., et al. 2018, MNRAS, 477, 1206
  * Navarro et al. (2004) Navarro, J. F., Abadi, M. G., & Steinmetz, M. 2004, ApJ, 613, L41
  * Nelson et al. (2018) Nelson, D., Pillepich, A., Springel, V., et al. 2018, MNRAS, 475, 624
  * Nelson et al. (2019) Nelson, D., Springel, V., Pillepich, A., et al. 2019, Computational Astrophysics and Cosmology, 6, 2
  * Pichon et al. (2011) Pichon, C., Pogosyan, D., Kimm, T., et al. 2011, MNRAS, 418, 2493
  * Pillepich et al. (2018) Pillepich, A., Nelson, D., Hernquist, L., et al. 2018, MNRAS, 475, 648
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A1
  * Press et al. (1992) Press, W. H., Teukolsky, S. A., Vetterling, W. T., et al. 1992, Numerical Recipes, (Cambridge: Cambridge University Press), 2nd ed.
  * Rodriguez-Gomez et al. (2015) Rodriguez-Gomez, V., Genel, S., Vogelsberger, M., et al. 2015, MNRAS, 449, 49
  * Romanowsky et al. (2003) Romanowsky, A. J., Douglas, N. G., Arnaboldi, M., et al. 2003, Science, 301, 1696
  * Schmidt et al. (1997) Schmidt, K.-H., Bohm, P., & Elsasser, H. 1997, Astronomische Nachrichten, 318, 81
  * Soussana et al. (2020) Soussana, A., Chisari, N. E., Codis, S., et al. 2020, MNRAS, 492, 4268
  * Springel et al. (2001) Springel, V., White, S. D. M., Tormen, G., et al. 2001, MNRAS, 328, 726
  * Springel et al. (2018) Springel, V., Pakmor, R., Pillepich, A., et al. 2018, MNRAS, 475, 676
  * Springel (2010) Springel, V. 2010, ARA&A, 48, 391. doi:10.1146/annurev-astro-081309-130914
  * Taruya & Okumura (2020) Taruya, A. & Okumura, T. 2020, ApJ, 891, L42.
  * Tempel et al. (2014) Tempel, E., Tamm, A., Gramann, M., et al. 2014, A&A, 566, A1
  * Tenneti et al. (2014) Tenneti, A., Mandelbaum, R., Di Matteo, T., et al. 2014, MNRAS, 441, 470
  * Tenneti et al. (2017) Tenneti, A., Gnedin, N. Y., & Feng, Y. 2017, ApJ, 834, 169.
  * Trowland et al. (2013) Trowland, H. E., Lewis, G. F., & Bland-Hawthorn, J. 2013, ApJ, 762, 72
  * Trujillo et al. (2006) Trujillo, I., Carretero, C., & Patiri, S. G. 2006, ApJ, 640, L111
  * Velliscig et al. (2015) Velliscig, M., Cacciato, M., Schaye, J., et al. 2015, MNRAS, 453, 721
  * Wang et al. (2014) Wang, X., Szalay, A., Aragón-Calvo, M. A., et al. 2014, ApJ, 793, 58
  * Wang & Kang (2017) Wang, P. & Kang, X. 2017, MNRAS, 468, L123
  * Wang et al. (2018) Wang, P., Guo, Q., Kang, X., et al. 2018, ApJ, 866, 138
  * Wang & Kang (2018) Wang, P., & Kang, X. 2018, MNRAS, 473, 1562
  * Weinberger et al. (2017) Weinberger, R., Springel, V., Hernquist, L., et al. 2017, MNRAS, 465, 3291
  * Welker et al. (2020) Welker, C., Bland-Hawthorn, J., Van de Sande, J., et al. 2020, MNRAS, 491, 2864
  * White (1984) White, S. D. M. 1984, ApJ, 286, 38
  * Zjupa & Springel (2017) Zjupa, J. & Springel, V. 2017, MNRAS, 466, 1625.

Figure 1: Mean values of the cosines of the angles of the galaxy stellar, cold
gas and hot gas spin axes from their DM spin axes (top, middle and bottom
panels, respectively), at $z=0$, averaged over each of the four controlled
samples classified by the latest merger epochs, $a(z_{m})$. The errorbars are
not visible due to their tiny sizes. The earlier the galaxies experience their
latest mergers, the weaker the spin alignments between their baryon and DM
components become.

Figure 2: Mean values of the galaxy total masses averaged over each of the
four controlled samples for three different cases of the particle number cuts:
$n_{s},n_{d}\geq 300$ (top panel); $n_{cg},n_{d}\geq 300$ (middle panel);
$n_{hg},n_{d}\geq 300$ (bottom panel).

Figure 3: Mean values of the environmental density contrasts averaged over
each of the four controlled samples for the three different cases of the
particle number cuts.

Figure 4: Mean absolute values of the cosines of the angles of the galaxy
stellar, cold gas, and hot gas spin vectors of the galaxies from the major
principal directions of the local tidal tensors (top, middle, and bottom
panels, respectively) averaged over each of the four controlled
$a(z_{m})$-selected samples. The earlier the galaxies experience their latest
mergers, the stronger the peculiar tidal connections of their stellar
components become.

Figure 5: Mean absolute values of the cosines of the angles of the galaxy DM
spin vectors from the major, intermediate and minor principal directions of
the local tidal tensors (top, middle and bottom panels, respectively) averaged
over each of the four controlled $a(z_{m})$-selected samples. No strong
dependence of the alignment strength between the galaxy DM spins and the
principal axes of the local tidal fields on the latest merger epochs.

Figure 6: Same as Figure 2 but for four different cases of the particle number
cuts: $n_{s}\geq 300$ (top panel); $n_{cg}\geq 300$ (second panel from the
top); $n_{hg}\geq 300$ (third panel from the top); $n_{d}\geq 300$ (bottom
panel).

Figure 7: Same as Figure 3 but for four different cases of the particle number
cuts: $n_{s}\geq 300$ (top panel); $n_{cg}\geq 300$ (second panel from the
top); $n_{hg}\geq 300$ (third panel from the top); $n_{d}\geq 300$ (bottom
panel).

Figure 8: Mean values of the dot products between the spin and shape vectors
of the galaxy stellar, cold gas, hot gas and DM components (top, second from
the top, third from the top and bottom panels, respectively), averaged over
each of the four controlled $a(z_{m})$-selected samples. In the top and bottom
panels, the errorbars are invisible due to their tiny sizes. The earlier the
latest mergers occur, the stronger (the weaker) the alignments between the
galaxy stellar (DM) spin and shape vectors become.

Figure 9: Mean absolute values of the dot products of the galaxy stellar, cold
gas, hot gas, DM shape vectors from the major principal directions of the
local tidal tensors (top, second from the top, third from the top and bottom
panels, respectively), averaged over each of the four controlled
$a(z_{m})$-selected samples. The earlier the latest mergers occur, the
stronger alignments the galaxies exhibit between their spin and shape vectors.

Figure 10: Mean values of the triaxiality of the galaxy stellar, cold gas, hot
gas and DM components averaged over the four controlled $a(z_{m})$-selected
samples. The errorbars in the top two panels are invisible due to their tiny
sizes. The earlier the latest mergers occur, the less triaxial shapes the
galaxies have.

Figure 11: Same as Figure 10 but of the sphericity. The errorbars in the
second panel from the top are invisible due to their tiny sizes. The earlier
the latest mergers occur, the more spherical shapes the galaxies have.