# Evolution of topological states in magnetic materials by symmetry-breaking

Feng Tang<EMAIL_ADDRESS>Xiangang Wan  National Laboratory of Solid
State Microstructures and School of Physics, Nanjing University, Nanjing
210093, China and Collaborative Innovation Center of Advanced Microstructures,
Nanjing University, Nanjing 210093, China

###### Abstract

The magnetism-controllable band topology renders magnetic topological
materials as one of the most promising candidates for next-generation
electronic devices. Here we first construct three datasets allowing
ascertaining the evolutions of topological states (with all enforced band
crossings identified) using symmetry-indicators in the 1651 magnetic space
groups. We then perform high-throughput investigations based on 1267
stoichiometric magnetic materials ever-experimentally synthesized and the
three datasets to reveal a hierarchy of topological states by symmetry-
breaking, along all ergodic and continuous paths of symmetry-breaking
(preserving the translation symmetry) from the parent magnetic space group to
the trivial group. The results are expected to aid experimentalists in
selecting feasible and appropriate means to tune topological states towards
realistic applications in new paradigm of memory device by topological state
switching.

## Introduction

The past nearly two decades have witnessed tremendous advancements of
symmetry-protected topological phases and topological materials Hasan and Kane
(2010); Qi and Zhang (2011); Chiu  _et al._ (2016); Ando and Fu (2015);
Armitage  _et al._ (2018) reshaping our fundamental understanding on the
electronic properties in solids. In the well-established paradigm of studying
topological phases in condensed matter, one usually utilizes the
symmetry/symmetries to classify the protected topological phases. Very
recently, topological quantum chemistry (TQC) Bradlyn  _et al._ (2017); Elcoro
_et al._ (2021) or symmetry-indicators (SIs) Po  _et al._ (2017); Watanabe
_et al._ (2018), have been applied in discovering topological nonmagnetic
Zhang  _et al._ (2019); Vergniory  _et al._ (2019); Tang  _et al._ (2019a);
Vergniory  _et al._ (2022)/magnetic Xu  _et al._ (2020)/superconducting Tang
_et al._ (2022) materials routinely and efficiently using the first-principles
calculated high-symmetry point (HSP) symmetry-data with respect to the space
groups/magnetic space groups (MSGs) of the pristine materials, revealing the
ubiquitous band topology, protected by MSG symmetry. Following the
conventional paradigm, finding or designing more symmetries beyond MSG
symmetry, such as spin-space group symmetry Liu  _et al._ (2022), dual
symmetry Fruchart  _et al._ (2020) and gauge-field induced projective symmetry
Chen  _et al._ (2023), is anticipated to unveil more topological phases, as an
on-going research theme in the field of topological materials.

On the other hand, symmetry-breaking can also be utilized in creating new
topological phase. For example, a higher-order band topology is induced by
applying strain on SnTe to break the mirror symmetry which originally protects
a mirror-Chern topological insulator phase Schindler  _et al._ (2018). Quantum
anomalous Hall effect was realized through gapping the time-reversal symmetry
protected Dirac fermion by magnetic dopants Yu  _et al._ (2010); Chang  _et
al._ (2013). Unconventional fermions can emerge in crystals which break the
Poincar$\acute{\mathrm{e}}$ symmetry Bradlyn  _et al._ (2016), and so on.
Hence, symmetry-breaking could introduce fruitful topological phases and can
act as a versatile knob to control topological states since the symmetry-
breaking can be easily-manipulated.

Furthermore, the advanced techniques on ultrafast light and high magnetic
fields have assisted experimentalists recently to stimulate the pristine
crystals to tune the topological states. The topological state can be
drastically modified leading to giant quantum responses which have been
observed in realistic materials, such as the colossal angular
magnetoresistance in ferrimagnetic Mn3Si2Te6 Seo  _et al._ (2021); Zhang  _et
al._ (2022); Ni  _et al._ (2021) and the giant anomalous Hall conductivity in
Heusler Co2MnAl Li  _et al._ (2020), where the topological states are
controlled by rotating the magnetic moments. Noticing that the magnetic
structure can be manipulated by magnetic fields to lower the symmetry Tokura
_et al._ (2019); Bernevig  _et al._ (2022), here we focus on the magnetic
materials listed in MAGNDATA MT- (a), which have been experimentally
synthesized and whose magnetic structures have already been deduced from
neutron-scattering experiments, and aim at revealing all topological states by
symmetry-breaking using SIs Po  _et al._ (2017); Tang  _et al._ (2019b);
Watanabe  _et al._ (2018) in these materials. The results are expected to
guide experimentalists to realize more magnetic topological materials with
unprecedentedly facsinating properties driven by the change of topological
phase.

Note that though concrete symmetry-breakings are not exhaustible, we can
classify symmetry-breakings by the resulting subgroups so that various
meticulous concrete symmetry-breakings corresponding to one subgroup, are
anticipated to share a common topological diagnosing prediction, since only
the transformation properties of HSP wavefunctions account Po  _et al._
(2017); Tang  _et al._ (2019b); Watanabe  _et al._ (2018). In this work, we
first construct three datasets based on the 1651 MSGs and their subgroups,
suitable to incorporate the symmetry-breaking to the conventional diagnostic
scheme of band topology based on SIs, and then apply them to 1267 magnetic
materials in MAGNDATA MT- (a) combined with first-principles calculations. The
three datasets will be described along with the description of the work flow
below.

## Work flow

The work flow is schematically shown in Fig. 1A. The first step is on the
first-principles calculations for the materials structures in the database MT-
(a) and the second step is to consider the symmetry-breaking. These two steps
utilize two datasets we constructed: dataset 1 named: “All_t-subgroups” and
dataset 2 named: “All_atomic_insulator_basis_sets”. In dataset 1, we exhaust
all translationengleiche subgroups (t-subgroups) for each of the 1651 MSGs,
which characterize symmetry-breaking in a general sense. Here, t-subgroup
means that the translation symmetry is still preserved and only the point
group symmetry is broken. To realize the t-subgroups, rotating magnetic
moments for magnetic materials is one feasible way, a strategy by which is
described in SM using Mn3Si2Te6 Seo  _et al._ (2021); Zhang  _et al._ (2022);
Ni  _et al._ (2021) as an example. The exhaustive list of all t-subgroups is
expected to be applied in various fields, not limited to the topological
materials here. For example, by dataset 1, we can find supergroups of a given
MSG. To realize a material with a small and nonvanishing band splitting which
could induce giant Berry curvatures in some MSGs, we can first target
materials with a higher MSG symmetry (namely, a supergroup) which enforces
band degeneracy (for which the band splitting is vanishing) and then apply
symmetry-breaking to materials crystallized in such supergroup to introduce a
gap on the originally degenerate bands with the gap proportional to the
strength of the symmetry-breaking which can be controllable. By dataset 1, all
subgroups with a given translation symmetry-breaking can also be identified
once the maximal subgroup for the translation symmetry-breaking is identified
Teng  _et al._ (2022); Zhou  _et al._ (see Supplementary Materials MT- (b)
(SM) for details). It is also worth mentioning that, in the materials
database, such as the nonmagnetic materials database, Inorganic Crystal
Structure Database (ICSD) Hellenbrandt (2004) and the magnetic materials
database, MAGNDATA MT- (a), the positions of atoms per unit cell are given
based on Wyckoff positions MT- (c) classified by the MSG. Then more
crystallographic symmetries might appear for some material, once the tolerance
of judging whether the structure is invariant by a candidate symmetry
operation, is relaxed. Hence, the corresponding identified MSG can be a
supergroup of the original one. The symmetries of the complement of the
original MSG in the supergroup can be regarded as one type of hidden symmetry,
which has attracted intensive interest very recently Fruchart  _et al._
(2020); Liu  _et al._ (2022); Chen  _et al._ (2023). The t-subgroups provided
here can be applied in finding such hidden symmetry by directly testing the
possible supergroups on the given structure.

After the convergence for some magnetic material is reached in the first-
principles calculation based on density functional theory in the first step
(also requiring that the resulting MSG reproduces that in MAGNDATA MT- (a),
called parent MSG hereafter), the related HSP symmetry data can be computed
based on the little groups for HSPs. All related data on the HSPs and the
little groups are all provided in dataset 2. From the calculated HSP symmetry-
data for the parent MSG, the HSP symmetry-data with respect to each t-subgroup
can be found using the compatibility relations between the HSPs of the
t-subgroup and the parent MSG, provided in dataset 1. In this way, additional
realistic calculations considering symmetry-breaking need not to be performed.
The topological classification for some t-subgroup can then be obtained by the
HSP symmetry-data with respect to the t-subgroup Po  _et al._ (2017); Tang
_et al._ (2019b), which can be applicable to any concrete symmetry-breaking as
along as it leads to the same t-subgroup. Note that we require that the HSP
symmetry-data for the t-subgroup can be induced from those for the parent MSG,
which is a reasonable approximation (see SM). In dataset 2, we also provide
the atomic insulator basis set Po  _et al._ (2017); Tang  _et al._ (2019b);
Watanabe  _et al._ (2018) of each MSG classifying any material into one of
three cases Tang  _et al._ (2019b), case I, II or III, of which case II/case
III definitely corresponds to a topological material (see SM for details).
Note that the t-subgroup is still an MSG, and thus we need not to construct
the atomic insulator basis set for the t-subgroup as long as we fix the
convention of the t-subgroup as that adopted in dataset 2.

In addition, to characterize the evolution of topological states, we identify
all continuous paths for the t-subgroups. We assign an integer to each
symmetry-breaking pattern (corresponding to a subgroup of the point group of
the parent MSG, see SM for details). We require that 0 represents that all
point operations in the parent MSG are preserved while 1 represents that only
identity operation ($E$) is preserved. The the continuous path can be written
in the form of $0\rightarrow i\rightarrow j\rightarrow\ldots\rightarrow 1$
where for all adjacent pair $i\rightarrow j$, the point operations for $i$
contain those for $j$, and there exists no other symmetry-breaking pattern
between $i$ and $j$. Note that each symmetry-breaking pattern uniquely
corresponds to one t-subgroup while there might exist several symmetry-
breaking patterns corresponding to a given t-subgroup. We find that the
maximal number of such continuous paths from 0 to 1 can be 8476. See an
example in Fig. 1A for MSG 12.62 (in the Belov-Neronova-Smirnova notation
Belov  _et al._ (1957)), where all t-subgroups and the corresponding symmetry-
breaking patterns are shown in the red circles of a tree-like plot. There are
in total 3 continuous paths from 0 to 1:

$0\rightarrow 2\rightarrow 1,0\rightarrow 3\rightarrow 1,0\rightarrow
4\rightarrow 1$

in terms of symmetry-breaking patterns, and $12.62\rightarrow 2.4\rightarrow
1.1,12.62\rightarrow 5.15\rightarrow 1.1,12.62\rightarrow 8.34\rightarrow 1.1$
in terms of t-subgroups. As listed in dataset 1, the point operations for the
symmetry-breaking patterns 0, 1, 2, 3, 4 are, $\\{E,I,\Theta C_{2z},\Theta
I\\}$, $\\{E\\}$, $\\{E,I\\}$,$\\{E,\Theta I\\}$ and
$\\{E,\Theta\sigma_{z}\\}$ ($I$ is spatial inversion, $\Theta$ is time-
reversal, $C_{2z}$ is 2-fold rotation around $z$-axis and $\sigma_{z}$ is a
mirror operation $=IC_{2z}$), respectively. Once the topological states are
identified for all t-subgroups, we can then obtain the evolution of
topological states along each continuous path.

The last step is devoted to the identification of all symmetry-enforced band
crossings (BCs) in case III. Note that in the earlier applications of SIs/TQC
to the high-throughput discovery of topological materials Zhang  _et al._
(2019); Vergniory  _et al._ (2019); Tang  _et al._ (2019a); Vergniory  _et
al._ (2022); Xu  _et al._ (2020), the characteristics of the enforced BCs are
not all identified. Very recently, there have been much effort on the complete
classification of all the BCs based on $k\cdot p$ models Yu  _et al._ (2022)
or combing $k\cdot p$ models and compatibility relations Tang and Wan (2022)
for the 230 space groups or the 1651 MSGs, respectively. Interestingly, the
expansion order of $k\cdot p$ model around the BC could have to be very large
(6 at most) Tang and Wan (2022) to capture the compatibility relations
required nodal structure. Then varying the expansion order introduces a
hierarchy of nodal structures around which tiny energy gaps could lead to
large Berry curvatures, which has been verified in CoSi experimentally Guo
_et al._ (2022). To this end, we provide dataset 3 (named
“Enforced_band_crossings”), where we list all possible enforced BCs for the
1651 MSGs, the compatibility relations required nodal structures and the
$k\cdot p$ models, and how to detect the enforced BCs in high-symmetry lines
(HSLs)/high-symmetry planes (HSPLs) by the HSP symmetry-data of the HSPs
residing in the HSLs/HSPLs. Using dataset 3, the enforced BCs by HSP symmetry-
data and their evolutions in the investigated magnetic materials are all
detected in this work.

Figure 1: Work flow of investigating the evolution of topological states. (A) The first step is performing the first-principles calculations on each selected magnetic material, and then the HSP symmetry-data with respect to the parent MSG are computed based on dataset 2, when the converged magnetic structure fulfills the parent MSG. Then by dataset 1, all HSP symmetry-data for all t-subgroups can be obtained. Combining the HSP symmetry-data and atomic insulator basis set (shown in dataset 2) for each t-subgroup, we then quickly classify the material in the t-subgroup into case I, II or III Tang  _et al._ (2019b). For case III, we obtain all enforced BCs by scanning all HSPs/HSLs/HSPLs in the Brillouin zone by dataset 3. (B) shows a case where case I can evolve to case II/III: The topmost level (indicated by the dashed ellipse) participating in the computation of the HSP symmetry-data, might be protected by some hidden symmetry and then carries two identical two-dimensional irreps (represented by two gray dots) at some HSP. By symmetry-breaking, the two-dimensional irrep splits to two different one-dimensional irreps, represented by red and blue dots, respectively. Consider two bands are filled. there is only one possibility of HSP symmetry-data before symmetry-breaking (one gray dot). However, by symmetry-breaking, there are three possibilities of irreps chosen to be occupied. As shown in the two panels, the original phase in case I can evolve to case I and case II/III, respectively. (C) demonstrates that the normal metal can be tuned to be a Kramers-Weyl semimetal Chang  _et al._ (2018) by symmetry-breaking: In the normal metal, energy bands are subject to two-fold Kramers degeneracy by the symmetry of combination of time-reversal and spatial inversion ($\Theta I$) with the filling an odd integer. Breaking the $\Theta I$ symmetry, the energy bands would split and only be degenerate at HSPs, where the Weyl points are resided. ID: N | ID: N | ID: N | ID: N | ID: N | ID: N | ID: N | ID: N | ID: N | ID: N
---|---|---|---|---|---|---|---|---|---
0.108: 9 | 0.109: 5 | 0.126: 5 | 0.140: 10 | 0.149: 8 | 0.174: 2 | 0.177: 10 | 0.186: 16 | 0.194: 5 | 0.199: 8
0.200: 10 | 0.203: 2 | 0.207: 4 | 0.227: 18 | 0.228: 9 | 0.236: 21 | 0.27: 10 | 0.273: 14 | 0.274: 5 | 0.275: 9
0.276: 8 | 0.279: 5 | 0.280: 8 | 0.286: 34 | 0.320: 22 | 0.321: 5 | 0.325: 13 | 0.327: 8 | 0.365: 5 | 0.367: 2
0.374: 8 | 0.377: 11 | 0.378: 5 | 0.395: 8 | 0.396: 8 | 0.402: 8 | 0.407: 5 | 0.408: 8 | 0.414: 11 | 0.415: 2
0.436: 11 | 0.445: 5 | 0.454: 46 | 0.461: 11 | 0.473: 2 | 0.486: 5 | 0.487: 1 | 0.495: 2 | 0.496: 2 | 0.497: 2
0.512: 3 | 0.561: 8 | 0.566: 5 | 0.593: 21 | 0.594: 21 | 0.595: 17 | 0.596: 21 | 0.599: 5 | 0.600: 5 | 0.603: 5
0.604: 5 | 0.605: 5 | 0.606: 5 | 0.609: 2 | 0.613: 18 | 0.614: 18 | 0.615: 18 | 0.616: 2 | 0.623: 5 | 0.625: 22
0.641: 2 | 0.656: 2 | 0.657: 2 | 0.662: 8 | 0.663: 8 | 0.664: 2 | 0.665: 16 | 0.681: 6 | 0.684: 5 | 0.699: 8
0.702: 2 | 0.703: 2 | 0.706: 5 | 0.707: 5 | 0.709: 8 | 0.711: 8 | 0.732: 8 | 0.737: 26 | 0.738: 26 | 0.74: 15
0.747: 2 | 0.771: 5 | 0.772: 8 | 0.773: 2 | 0.774: 5 | 0.776: 8 | 0.777: 8 | 0.778: 5 | 0.779: 5 | 0.780: 5
0.80: 22 | 0.81: 5 | 1.0.11: 2 | 1.0.12: 8 | 1.0.29: 2 | 1.0.43: 15 | 1.102: 54 | 1.103: 16 | 1.104: 16 | 1.110: 16
1.130: 16 | 1.131: 16 | 1.132: 16 | 1.139: 16 | 1.143: 99 | 1.146: 46 | 1.150: 16 | 1.16: 16 | 1.160: 57 | 1.162: 46
1.179: 16 | 1.187: 67 | 1.188: 16 | 1.207: 16 | 1.208: 46 | 1.21: 57 | 1.215: 27 | 1.222: 16 | 1.223: 16 | 1.251: 46
1.252: 16 | 1.253: 27 | 1.254: 27 | 1.255: 16 | 1.261: 35 | 1.290: 16 | 1.291: 46 | 1.296: 16 | 1.305: 16 | 1.308: 16
1.338: 46 | 1.369: 16 | 1.384: 27 | 1.398: 16 | 1.399: 16 | 1.400: 27 | 1.419: 21 | 1.421: 91 | 1.422: 42 | 1.423: 35
1.425: 27 | 1.427: 46 | 1.428: 46 | 1.442: 46 | 1.446: 16 | 1.453: 132 | 1.458: 36 | 1.460: 16 | 1.461: 46 | 1.468: 132
1.469: 132 | 1.475: 16 | 1.479: 19 | 1.487: 46 | 1.488: 27 | 1.489: 46 | 1.490: 27 | 1.491: 132 | 1.492: 132 | 1.493: 27
1.494: 46 | 1.495: 132 | 1.496: 132 | 1.497: 5 | 1.508: 16 | 1.512: 27 | 1.513: 113 | 1.514: 27 | 1.516: 16 | 1.52: 16
1.530: 46 | 1.531: 16 | 1.532: 46 | 1.534: 16 | 1.536: 46 | 1.537: 35 | 1.549: 70 | 1.555: 16 | 1.558: 16 | 1.559: 16
1.568: 5 | 1.575: 16 | 1.585: 16 | 1.623: 5 | 1.628: 27 | 1.635: 16 | 1.636: 27 | 1.637: 27 | 1.638: 27 | 1.639: 27
1.640: 27 | 1.80: 10 | 1.81: 46 | 1.82: 10 | 1.85: 22 | 1.87: 27 | 1.88: 16 | 2.1: 16 | 2.10: 2 | 2.11: 5
2.12: 5 | 2.13: 46 | 2.14: 27 | 2.19: 18 | 2.26: 21 | 2.28: 5 | 2.31: 16 | 2.48: 38 | 2.5: 6 | 2.54: 5
2.57: 8 | 2.59: 2 | 2.65: 21 | 2.70: 2 | 2.77: 72 | 2.81: 8 | 2.82: 8 | 2.83: 9 | 2.84: 8 | 3.10: 134
3.11: 134 | 3.12: 30 | 3.6: 25 | 3.7: 30 | 3.8: 134 | 3.9: 186 |  |  |  |
Table 1: The 236 magnetic topological materials predicted to be topological
for at least 3 values of $U$ at the intrinsic filling. ID denotes the ID or
label in MAGNDATA MT- (a) and N denotes the number of t-subgroups with respect
to which the corresponding material is topological for the different choices
of values of $U$. For example, ID = 1.308 in the last column uniquely
corresponds to the magnetic topological insulator MnBi2Te4 Bernevig  _et al._
(2022).
Figure 2: Statistics of identified magnetic topological materials in this
work. The filling is tuned up to a difference by 4. For $|\Delta\nu|\leq
0,1,2,3$ or $4$, four numbers are counted: two are for the pristine magnetic
topological materials (for which only the parent MSGs are exploited) and two
are related with all t-subgroups (including the parent MSGs).

## High-throughput investigations on 1267 magnetic materials

Next we discuss the high-throughput first-principles investigations on the
magnetic materials as listed in MAGNDATA MT- (a), all synthesized
experimentally whose magnetic structures are measured. Of the 1721 magnetic
materials in total which we collected at the time of initiating this work, we
first filter out materials with incommensurate magnetic ordering, without a
definite MSG, and further requiring that the atoms fully occupy their sites,
the lattice parameters are compatible with the MSG and the numbers of ions are
compatible with the chemical formula, we finally obtain 1267 “high-quality”
magnetic materials as our starting point for subsequent first-principles
calculations and topological classification following the work flow shown in
Fig. 1A. The experimentally identified MSGs MT- (a) are set as the parent
MSGs. The electron correlation is considered by choosing various values of
Hubbard $U$, and thus we have 5883 jobs in total for the 1267 materials (one
material with one value of $U$ is regarded as one job). Finally, 5062 jobs
reach convergence in the first-principles calculations. However, the converged
magnetic structure might break the original MSG symmetry. We then have 4012
jobs with successfully computed HSP symmetry-data corresponding to 295 MSGs
and 1013 magnetic materials. We list all these 295 MSGs in Table S18 of SM
where the number of t-subgroups, the number of materials in each MSG and the
number of jobs are also listed. For these jobs, we compute the representations
for all the energy levels at the HSPs for the parent MSG by the first-
principles calculated wavefunctions. In order to detect all topological bands
around the Fermi level, we vary the electron filling to be
$\nu=\nu_{0},\nu_{0}\pm 1,\nu_{0}\pm 2,\nu_{0}\pm 3,\nu_{0}\pm 4$ ($\nu_{0}$
is the intrinsic filling, which is the number of valence electrons per
primitive unit cell). Tuning filling might also lead to different topological
phases, as has realized in a topological phase change transistor by
electrostatic gating Chen  _et al._ (2022). In total, for each filling, we
collect 150882 sets of HSP symmetry-data considering all the 4012 jobs and all
t-subgroups, all listed in SM. These HSP symmetry-data are then exploited to
conduct all subsequent topological classifications including the
identification of all enforced BCs.

## Magnetic topological materials statistics

The statistics of the numbers of the identified magnetic topological materials
in this work is shown in Fig. 2, which monotonically increase and finally
almost approach constants with the variations of fillings. Considering the
intrinsic filling first, 392 magnetic materials are identified to be
topological, belonging to case II or III, and 236 are predicted to be
topological for at least 3 different values of Hubbard $U$, whose ID,s in
MAGNDATA MT- (a) are listed in Table 1. The proportion of magnetic topological
materials is consistent with that revealed in Ref. Xu  _et al._ (2020). The
number of topological materials is gradually increased to 945 when the
fillings satisfying $|\Delta\nu|\leq 4$ ($\Delta\nu=\nu-\nu_{0}$), indicating
that more than 90% of the investigated magnetic materials can be topological.

For all considered fillings, we verify that, when all the topmost levels are
fully occupied for all HSPs in the parent MSG, the evolution of topological
states in all these materials obey the intuitive picture: case I changes to
case I, case II changes to case I/II and case III changes to case I/II/III for
any adjacent pair $i\rightarrow j$ in each continuous path by symmetry-
breaking. To what extent the band topology by symmetry-breaking can be
preserved could be meaningful in the optimization of topological materials,
which might break some symmetry. In Table 1, we then show the number of
t-subgroups in which the nontrivial topology is preserved for at least three
values of $U$ for the listed magnetic materials. Interestingly, for other
materials, it might occur that case I can changes to case II/III and case II
can changes to case III: When the topmost level at some HSP is not fully
occupied and contains more than one (co-)irreps (which might be caused by some
hidden symmetry), one need to choose one or several (co-)irreps to be occupied
while a lowered symmetry can allow different (co-)irreps to be chosen, as
shown in Fig. 1B leading to the counterintuitive result. Note that In Fig. 1C,
we also show a possibility that a normal metal phase, where the filling is an
odd number while the bands are subject to a Kramers degeneracy in the whole
Brillouin zone, changes to a Kramers-Weyl semimetal phase Chang  _et al._
(2018) by symmetry-breaking. Considering all t-subgroups, there are 1003
magnetic materials which can be topological at some filling with respect to
some t-subgroup and for some value of $U$. We list all these 1003 materials in
Tables S19-S94 of SM, where the corresponding fillings for case II, case III
and case II/III only when considering symmetry-breaking are listed for each of
these material. We also assemble all results for all the 5883 jobs in Table
S95-S287 of SM, where the calculated magnetic moments, the detailed
topological properties, the plots of band and density of states (DOS) can be
found.

We manually select 9 magnetic topological materials whose electronic band
plots are depicted in Fig. 3 and describe the results by symmetry-breaking in
these materials. These materials own either a noticeable band gap or enforced
BC almost at the Fermi level. Consider the intrinsic filling. The materials in
Fig. 3 are topological for at least four values of $U$ other than Sr2TbIrO6,
for which only two jobs ($U=$0, 4 eV) reach convergence. The topological
classifications for Sr2TbIrO6 with $U=0$ and 4 eV are identical, with respect
to all t-subgroups (including the parent MSG): for t-subgroup MSG 2.4, SI $=$
$(0,0,0,2)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$
Watanabe  _et al._ (2018) while for other t-subgroups, the classifications are
case I. Note that different values of $U$ might correspond to different
topological predictions (case II/III) in UAsS and Ba3CoIr2O9. For others,
different values of $U$ with topological prediction give either case II or
case III. By symmetry-breaking, the nontrivial band topology in these
materials can be maintained, transformed or trivialized. For example, 21 of
the 35 t-subgroups for UAsS ($U$ $=$ 0, 2, 4, 6 eV) give case II/case III
prediction, indicating the robustness of nontrivial band topology against
symmetry-breaking. For CaMnSi ($U$ $=$ 0, 1, 2, 3 eV), 5 of the 35 t-subgroups
give case II while the rest give case I. For CsMnF4 with $U=0,1,2,3$ eV, the
topological classifications share the same results of topological states with
respect to all t-subgroups. For this material, 8 of the 16 t-subgroups give
case III prediction and the rest give case I, and the enforced BCs include two
pairs of opposite-charged isoenergetic Weyl points and one Weyl nodal line for
the parent MSG. The Weyl points of each pair are related by inversion
symmetry. By symmetry-breaking, the original Weyl points/nodal line can be
gapped: For example, with respect to the t-subgroup MSG 6.18, the Weyl points
are all gapped while the Weyl nodal line still exists); With respective to the
t-subgroup 18.19, the original inversion related Weyl points are allowed to
own different energies.

In the following, we take HoB2 and CeMn2Si2 as examples to show the detailed
evolutions of topological states along the continuous paths.

Figure 3: Electronic band plots of nine manually selected magnetic topological
materials identified in this work. In all the band plots in this work, the
Fermi level is set to be 0 eV. The chemical formula, the ID in MAGNDATA MT-
(a), the MSG, the Hubbard $U$ are all provided in the head of each band plot.
Consider the intrinsic filling. Other than CsMnF4 in (B) and CeMn2Si2, which
are classified in case III, the rest are in case II. Sr2TbIrO6, UAsS, CaMnSi,
Ba3CoIr2O9, EuMg2Bi2, Mn2AlB2 and HoB2 own SI $=$
$(1)\in\mathbb{Z}_{2},(0,1)\in\mathbb{Z}_{2}\times\mathbb{Z}_{4},(1)\in\mathbb{Z}_{2},$$(1,0)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2},(1)\in\mathbb{Z}_{2},(1)\in\mathbb{Z}_{2},(0,0,2)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$,
respectively. For CsMnF4, the enforced BCs appear in
$(-w,0,0),(-w,0,\frac{1}{2}),(0,w,0),(0,w,\frac{1}{2}),(u,0,v)$. For CeMn2Si2,
the enforced BCs appear in
$(0,w,\frac{1}{2}),(w,w,\frac{1}{2}),(u,v,\frac{1}{2})$. The effect of
symmetry-breaking on the topological properties are described in the main
text. The topological properties with respect to all t-subgroups (including
the parent MSG) can be found in SM.
Figure 4: Evolution of topological states in HoB2 and CeMn2Si2. In the upper
panel of (A) are shown the band structure where all the bands are
nondegenerate and the plot of DOS. The continuous gaps corresponding to the
intrinsic filling $\nu_{0}$ and the fillings
$\nu_{0}-1,\nu_{0}+1,\nu_{0}+2,\nu_{0}+3$ are indicated by the regions in
yellow, cyan, orange, green and magenta, respectively. The lower panel of (A)
shows the tree-like plots for the topological state evolution corresponding to
the gaps indicated by the colors shown in the insets. The symmetry-breaking
patterns are shown: 0, 1, 2, 3 and 4 correspond to the t-subgroups 12.62, 1.1,
2.4, 5.15 and 8.34, respectively. Note that the peaks in the DOS plot
correspond to the two bands (the band indices are $\nu_{0}+2$ and $\nu_{0}+3$)
indicated by the white double-arrow and such quasi-flat bands are topological
(with SI $=$ $(1,1,3)$). (B) shows the band structures along two HSLs,
$(0,w,1/2)$ and $(w,w,1/2)$, where the orange and blue colors encode two
different co-irreps of the corresponding HSLs. Various BCs composed of these
two co-irreps can be found and the BCs indicated by the red arrows are deduced
by the HSP symmetry-data: Along these two HSLs are the HSPs Z
$(0,0,\frac{1}{2})$-R $(0,\frac{1}{2},\frac{1}{2})$-Z $(0,1,\frac{1}{2})$, and
Z $(0,0,\frac{1}{2})$-A $(-\frac{1}{2},-\frac{1}{2},\frac{1}{2})$-Z
$(-1,-1,\frac{1}{2})$, respectively. Note that in each of these HSLs, there
exists another BC related with the one indicated by the red arrow by an MSG
operation. (C) shows the tree-like plot for the evolution of the original
Dirac nodal lines in CeMn2Si2. In each circle, the integer denotes the
symmetry-breaking patten, and II stands for case II and WNL/DNL stands for
Weyl nodal line/Dirac nodal line in case III.

## Materials example: HoB2

We first show an example of topological magnetic material, a rare-earth
diboride, HoB2, classified to belong to case II. The ID in MAGNDATA MT- (a) is
0.616. HoB2 in the paramagnetic state is crystallized in a hexagonal lattice
(space group: $P6/mmm$) with a very simple crystal structure containing only
one chemical formula per primitive unit cell, while a ferromagnetic state
breaks the three-fold rotation symmetry, resulting in MSG 12.62, reported to
display a gigantic magnetocaloric effect de Castro  _et al._ (2020); Terada
_et al._ (2020). For MSG 12.62, all energy bands should be nondegenerate, then
BCs are not allowed to exist stably at the special $k$ points, as shown in
dataset 3, while it owns a nontrivial SI group Watanabe  _et al._ (2018),
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}$. Indeed, for all
t-subgroups, HoB2 can only be case I or case II as shown for different
fillings in the lower panel of Fig. 3A
($\nu=\nu_{0},\nu_{0}-1,\nu_{0}+1,\nu_{0}+2$ and $\nu_{0}+3$) which
characterize the evolution of topological states by the tree-like plots. The
fillings correspond to the gaps shown in the upper panel of Fig. 3A, indicated
by different colors. It is interesting to note that there are considerable
gaps for the filling $\nu_{0}+1$ (the region in orange in the upper panel of
Fig. 3A), the nontrivial topology of which is expected to bring about
noticeable protected boundary states. Such gaps share the same evolution of
topological states as those for another three fillings as shown in the lower
panel of Fig. 3A, where the colors shown in the inset correspond to the gaps
in the respective colors. Furthermore, the quasi-flat bands (whose band
indices are $\nu_{0}+2$ and $\nu_{0}+3$) are found to topological by comparing
the SIs in two tree-like plots of the lower panel of Fig. 3A. The coexistence
of magnetic order, topological quasi-flat bands and multiple topological gaps
around the Fermi level makes this material a fascinating platform to study
possible exotic quantum excitations.

## Materials example: CeMn2Si2

We then show another example for CeMn2Si2 whose ID in MAGNDATA is 1.490 MT-
(a) and MSG is 126.386. The Ce ion is found to own vanishing magnetism while
the magnitude of the magnetic moments in all Mn ions are 1.9 $\mu$B
experimentally Fernandez-Baca  _et al._ (1996). We chose the results for the
calculations in which all the values of $U$ on the Ce and Mn ions are $0$ eV,
in which the calculated magnetic moments reasonably reproduce those by
experiments. It is noted that the other results for the rest values of $U$
share similar topological properties discussed here. Consider the intrinsic
filling. In the parent MSG, CeMn2Si2 is predicted to be in case III and
furthermore, the HSP symmetry-data guarantee that the enforced BCs appear in
HSLs, $(0,w,\frac{1}{2})$ and $(w,w,\frac{1}{2})$ by dataset 3, as shown in
Fig. 3B, where these BCs are also verified from the representations of all the
$k$ points in a dense $k$ mesh in the HSLs. It is also found that the BCs in
these HSLs actually lie in a nodal line in an HSPL ($(u,v,\frac{1}{2})$) by
dataset 3, with a degeneracy 4, denoted to be Dirac nodal line. Then consider
the effect of the symmetry-breaking. These Dirac nodal lines can be gapped in
some symmetry-breaking patterns, resulting in a possibly-trivial insulator (in
case I) or a topological phase in case II. In other symmetry-breaking
patterns, the resulting topological phases are all in case III, and
furthermore, the guaranteed BC lies in either Dirac nodal line or Weyl nodal
line (whose degeneracy is two). We show the evolution of topological states
for this material in Fig. 3C. It is worth mentioning that the tree-like plot
in Fig. 3C only contains the symmetry-breaking patterns resulting in case
II/III, and the symmetry-breaking patterns following the symmetry-breaking
pattern in the end of the tree-like plot (15 and 57) all result in case I, a
possibly trivial topological phase.

## Conclusion and perspective

To conclude, a complete dataset of all t-subgroups for the 1651 MSGs and the
compatibility relations between the HSPs of each pair of the parent MSG and
its t-subgroup was constructed. The dataset can also be applied in studying
the symmetry-breaking of the superconducting pairing belonging to higher-
dimensional irreps of point groups Ono  _et al._ (2021); Tang  _et al._
(2022). The 1651 atomic insulator basis sets were also provided, which can be
applied to any magnetic/nonmagnetic material with the identified MSG,
experimentally MT- (a) or theoretically Frey  _et al._ (2020). The HSPs, their
little groups and the characters of (co-)irreps were explicitly provided in
the 1651 MSGs for both Bradley-Cracknell Bradley and Cracknell  and Bilbao MT-
(c) conventions, two frequently-adopted conventions. The topological
classification results for more than one thousand magnetic materials obtained
in this work can guide experimentalists to choose their interested material
and the way of breaking symmetry, which might pave the avenue to the
realization of highly-sensitive magnetism-controlled band topology and the
topological magnets with high operational temperature. They can also be used
as the training dataset to develop a simple-to-use heuristic chemical rule of
diagnosing band topology such as the topogivity Ma  _et al._ (2023) by
machine-learning. The enforced of BCs are identified by the HSP symmetry-data,
already producing a large number of BCs with various topological characters:
We have collected 179 and 2006 independent BCs at HSPs and lying in HSLs,
respectively, considering the intrinsic fillings. It should be noted that to
identify all BCs, we need to adopt a poor man’s strategy: a dense $k$-point
mesh should be chosen in HSLs/HSPLs and all the symmetry-data at the $k$
points in the mesh should be evaluated, which is left to the future work. Note
that the topological classifications for the t-subgroups here for magnetic
materials are irrelevant with the origin of the symmetry-breaking, so that the
results are applicable widely in a general sense. Since we identified all
continuous paths of t-subgroups by symmetry-breaking, the corresponding
evolution of topological states along some continuous path guarantees no
omission of topological states detected in a concrete process of symmetry-
breaking. Inversely, the topological state evolution might indicate the type
of symmetry-breaking. Other than useful in guiding the control of the
topological states, the results are expected to aid in the assessment on the
robustness of the nontrivial band topology against symmetry-breaking, which
could be useful in the optimization of topological materials or the transport
measurements which need an application of external stimuli.

The correlation of magnetism and nontrivial band topology with other ordering,
such as ferroelectric order Khomskii (2009), and novel crystal structure (e.g.
Kagom$\acute{\mathrm{e}}$ lattice), merits future studies by choosing suitable
candidates from our predicted magnetic topological materials. Lastly, a more
generic symmetry-breaking could include the translation symmetry breaking
enlarging the primitive unit cell and lead to a Brillouin zone folding (e.g. a
charge-density wave order Teng  _et al._ (2022); Zhou  _et al._ ), which is
not exhaustible, but can still apply the t-subgroups we obtained here after
firstly identifying the subgroup(s) by only considering the translation
symmetry-breaking, of which the details are described in SM. Another further
work that can be based on the t-subgroups exhaustively listed here is to
explore the splitting of each Wyckoff position for each MSG in the t-subgroup,
which might guide the design for approximate MSG symmetry simply by the atomic
positions once the Wyckoff position splitting can be ignored approximately.
The t-subgroups for the 1651 MSGs could also be utilized to tabulate novel
groups beyond the MSGs by homomorphism theorem. Besides, one can use the
t-subgroups (or supergroups) of MSG for the Kagom$\acute{\mathrm{e}}$ lattice,
Lieb lattice or other interesting lattice showing exotic physics properties to
find materials realizations Regnault  _et al._ (2022).

## Acknowledgments

We are very grateful for earlier collaborations on related topics with Ashvin
Vishwanath, Hoi Chun Po, Haruki Watanabe and Seishiro Ono. F.T. appreciates
insightful discussions with Wei Chen, Qun-Li Lei, Kai Li and Yang-Yang Lv. We
also thank very helpful suggestions from Ge Yao on high-performance computing.
Funding: F.T. was supported by National Natural Science Foundation of China
(NSFC) under Grant No. 12104215 and Young Elite Scientists Sponsorship Program
by China Association for Science and Technology. F.T. and X.W. were supported
by NSFC Grants No. 12188101, 11834006, 51721001, and 11790311, and the
excellent program at Nanjing University. X.W. also acknowledges the support
from the Tencent Foundation through the XPLORER PRIZE. Author contributions:
F.T. conceived and designed the project, and performed all calculations. All
authors contributed to the writing and editing of the manuscript. Competing
interests: None declared. Data and materials availability: All data are
available in the manuscript or the supplementary materials. The three datasets
constructed in this work can be found in:
https://box.nju.edu.cn/published/three-datasets/.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057 (2011).
  * Chiu  _et al._ (2016) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries,” Rev. Mod. Phys. 88, 035005 (2016).
  * Ando and Fu (2015) Y. Ando and L. Fu, “Topological crystalline insulators and topological superconductors: From concepts to materials,” Annu. Rev. Condens. Matter Phys. 6, 361–381 (2015).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).
  * Bradlyn  _et al._ (2017) B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, “Topological quantum chemistry,” Nature 547, 298–305 (2017).
  * Elcoro  _et al._ (2021) L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, “Magnetic topological quantum chemistry,” Nat. Commun. 12, 5965 (2021).
  * Po  _et al._ (2017) H. C. Po, A. Vishwanath, and H. Watanabe, “Symmetry-based indicators of band topology in the 230 space groups,” Nat. Commun. 8, 50 (2017).
  * Watanabe  _et al._ (2018) H. Watanabe, H. C. Po, and A. Vishwanath, “Structure and topology of band structures in the 1651 magnetic space groups,” Sci. Adv. 4, aat8685 (2018).
  * Zhang  _et al._ (2019) T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, “Catalogue of topological electronic materials,” Nature 566, 475–479 (2019).
  * Vergniory  _et al._ (2019) M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, “A complete catalogue of high-quality topological materials,” Nature 566, 480–485 (2019).
  * Tang  _et al._ (2019a) F. Tang, H. C. Po, A. Vishwanath, and X. Wan, “Comprehensive search for topological materials using symmetry indicators,” Nature 566, 486–489 (2019a).
  * Vergniory  _et al._ (2022) M. G. Vergniory, B. J. Wieder, L. Elcoro, S. S. P. Parkin, C. Felser, B. A. Bernevig, and N. Regnault, “All topological bands of all nonmagnetic stoichiometric materials,” Science 376, abg9094 (2022).
  * Xu  _et al._ (2020) Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, “High-throughput calculations of magnetic topological materials,” Nature 586, 702–707 (2020).
  * Tang  _et al._ (2022) F. Tang, S. Ono, X. Wan, and H. Watanabe, “High-Throughput Investigations of Topological and Nodal Superconductors,” Phys. Rev. Lett. 129, 027001 (2022).
  * Liu  _et al._ (2022) P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, “Spin-Group Symmetry in Magnetic Materials with Negligible Spin-Orbit Coupling,” Phys. Rev. X 12, 021016 (2022).
  * Fruchart  _et al._ (2020) M. Fruchart, Y. Zhou, and V. Vitelli, “Dualities and non-Abelian mechanics,” Nature 577, 636–640 (2020).
  * Chen  _et al._ (2023) Z. Y. Chen, Z. Zhang, S. A. Yang, and Y. X. Zhao, “Classification of time-reversal-invariant crystals with gauge structures,” Nat. Commun. 14, 743 (2023).
  * Schindler  _et al._ (2018) F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, “Higher-order topological insulators,” Sci. Adv. 4, eaat0346 (2018).
  * Yu  _et al._ (2010) R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, “Quantized Anomalous Hall Effect in Magnetic Topological Insulators,” Science 329, 61–64 (2010).
  * Chang  _et al._ (2013) C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, “Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator,” Science 340, 167–170 (2013).
  * Bradlyn  _et al._ (2016) B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, “Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals,” Science 353, aaf5037 (2016).
  * Seo  _et al._ (2021) J. Seo, C. De, H. Ha, J. E. Lee, S. Park, J. Park, Y. Skourski, E. S. Choi, B. Kim, G. Y. Cho, H. W. Yeom, S.-W. Cheong, J. H. Kim, B.-J. Yang, K. Kim, and J. S. Kim, “Colossal angular magnetoresistance in ferrimagnetic nodal-line semiconductors,” Nature 599, 576–581 (2021).
  * Zhang  _et al._ (2022) Y. Zhang, Y. Ni, H. Zhao, S. Hakani, F. Ye, L. DeLong, I. Kimchi, and G. Cao, “Control of chiral orbital currents in a colossal magnetoresistance material,” Nature 611, 467–472 (2022).
  * Ni  _et al._ (2021) Y. Ni, H. Zhao, Y. Zhang, B. Hu, I. Kimchi, and G. Cao, “Colossal magnetoresistance via avoiding fully polarized magnetization in the ferrimagnetic insulator Mn3Si2Te6,” Phys. Rev. B 103, L161105 (2021).
  * Li  _et al._ (2020) P. Li, J. Koo, W. Ning, J. Li, L. Miao, L. Min, Y. Zhu, Y. Wang, N. Alem, C.-X. Liu, Z. Mao, and B. Yan, “Giant room temperature anomalous Hall effect and tunable topology in a ferromagnetic topological semimetal Co2MnAl,” Nat. Commun. 11, 3476 (2020).
  * Tokura  _et al._ (2019) Y. Tokura, K. Yasuda, and A. Tsukazaki, “Magnetic topological insulators,” Nat. Rev. Phys. 1, 126–143 (2019).
  * Bernevig  _et al._ (2022) B. A. Bernevig, C. Felser, and H. Beidenkopf, “Progress and prospects in magnetic topological materials,” Nature 603, 41–51 (2022).
  * MT- (a) “http://webbdcrista1.ehu.es/magndata/,” (a).
  * Tang  _et al._ (2019b) F. Tang, H. C. Po, A. Vishwanath, and X. Wan, “Efficient topological materials discovery using symmetry indicators,” Nat. Phys. 15, 470–476 (2019b).
  * Teng  _et al._ (2022) X. Teng, L. Chen, F. Ye, E. Rosenberg, Z. Liu, J.-X. Yin, Y.-X. Jiang, J. S. Oh, M. Z. Hasan, K. J. Neubauer, B. Gao, Y. Xie, M. Hashimoto, D. Lu, C. Jozwiak, A. Bostwick, E. Rotenberg, R. J. Birgeneau, J.-H. Chu, M. Ying, and P. Dai, “Discovery of charge density wave in a kagome lattice antiferromagnet,” Nature 609, 490–495 (2022).
  * (32) H. Zhou, S. Yan, D. Fan, D. Wang, and X. Wan, “Magnetic interactions and possible structural distortion in kagome fege from first-principles study and symmetry analysis,” arXiv: 2211.15545 (2022) .
  * MT- (b) “See Supplementary Materials,” (b).
  * Hellenbrandt (2004) M. Hellenbrandt, “The inorganic crystal structure database (ICSD)-present and future,” Crystallography Reviews 10, 17–22 (2004).
  * MT- (c) “https://www.cryst.ehu.es/,” (c).
  * Belov  _et al._ (1957) N. V. Belov, N. N. Neronova, and T. S. Smirnova, “Shubnikov groups,” Sov. Phys. Crystallogr. 2, 311–322 (1957).
  * Yu  _et al._ (2022) Z.-M. Yu, Z. Zhang zna G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S. A. Yang, and Y. Yao, “Encyclopedia of emergent particles in three-dimensional crystals,” Sci. Bull. 67, 375–380 (2022).
  * Tang and Wan (2022) F. Tang and X. Wan, “Complete classification of band nodal structures and massless excitations,” Phys. Rev. B 105, 155156 (2022).
  * Guo  _et al._ (2022) C. Guo, L. Hu, C. Putzke, J. Diaz, X. Huang, K. Manna, F.-R. Fan, C. Shekhar, Y. Sun, C. Felser, C. Liu, B. A. Bernevig, and P. J. W. Moll, “Quasi-symmetry-protected topology in a semi-metal,” Nat. Phys. 18, 813–818 (2022).
  * Chang  _et al._ (2018) G. Chang, B. J. Wieder, F. Schindler, D. S. Sanchez, I. Belopolski, S.-M. Huang, B. Singh, D. Wu, T.-R. Chang, T. Neupert, S.-Y. Xu, H. Lin, and M. Z. Hasan, “Topological quantum properties of chiral crystals,” Nat. Mater. 17, 978–985 (2018).
  * Chen  _et al._ (2022) J. Chen, T. Zhang, J. Wang, L. Xu, Z. Lin, J. Liu, C. Wang, N. Zhang, S. P. Lau, W. Zhang, M. Chhowalla, and Y. Chai, “Topological phase change transistors based on tellurium Weyl semiconductor,” Sci. Adv. 8, abn3837 (2022).
  * de Castro  _et al._ (2020) P. B. de Castro, K. Terashima, T. D. Yamamoto, Z. Hou, S. Iwasaki, R. Matsumoto, S. Adachi, Y. Saito, P. Song, H. Takeya, and Y. Takano, “Machine-learning-guided discovery of the gigantic magnetocaloric effect in HoB2 near the hydrogen liquefaction temperature,” NPG Asia Mater. 12, 35 (2020).
  * Terada  _et al._ (2020) N. Terada, K. Terashima, P. B. de Castro, C. V. Colin, H. Mamiya, T. D. Yamamoto, H. Takeya, O. Sakai, Y. Takano, and H. Kitazawa, “Relationship between magnetic ordering and gigantic magnetocaloric effect in HoB2 studied by neutron diffraction experiment,” Phys. Rev. B 102, 094435 (2020).
  * Fernandez-Baca  _et al._ (1996) J. A. Fernandez-Baca, P. Hill, B. C. Chakoumakos, and N. Ali, “Neutron diffraction study of the magnetic structures of CeMn2Ge2 and CeMn2Si2,” J. Appl. Phys. 79, 5398 (1996).
  * Ono  _et al._ (2021) S. Ono, H. C. Po, and K. Shiozaki, “$\mathbb{Z}_{2}$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups,” Phys. Rev. Res. 3, 023086 (2021).
  * Frey  _et al._ (2020) N. C. Frey, M. K. Horton, J. M. Munro, S. M. Griffin, K. A. Persson, and V. B. Shenoy, “High-throughput search for magnetic and topological order in transition metal oxides,” Sci. Adv. 6, eabd1076 (2020).
  * (47) C. J. Bradley and A. P. Cracknell, “The Mathematical Theory of Symmetry in Solids (oxford university press, oxford, 1972),” .
  * Ma  _et al._ (2023) A. Ma, Y. Zhang, T. Christensen, H. C. Po, L. Jing, L. Fu, and M. Solja$\check{\mathrm{c}}$i$\acute{\mathrm{c}}$, “Topogivity: A Machine-Learned Chemical Rule for Discovering Topological Materials,” Nano Lett. 3, 772–778 (2023).
  * Khomskii (2009) D. Khomskii, “Classifying multiferroics: Mechanisms and effects,” Physics 2, 20 (2009).
  * Regnault  _et al._ (2022) N. Regnault, Y. Xu, M.-R. Li, D.-S. Ma, M. Jovanovic, A. Yazdani, S. S. P. Parkin, C. Felser, L. M. Schoop, N. P. Ong, R. J. Cava, L. Elcoro, Z.-D. Song, and B. A. Bernevig, “Catalogue of flat-band stoichiometric materials,” Nature 603, 824–828 (2022).