Topological state evolution by symmetry-breaking

TL;DR

This paper develops a systematic framework to analyze how topological states in magnetic and non-magnetic materials evolve under symmetry-breaking, using a tree-like graph representation, supported by high-throughput data on magnetic materials.

Contribution

It introduces a novel tree-like graph framework for topological state evolution under symmetry-breaking across all magnetic space groups, and applies it to a large dataset of magnetic materials.

Findings

Different symmetry-breaking pathways lead to various contractions of the topological state graph.

High-throughput analysis reveals a hierarchy of topological states along symmetry-breaking paths.

Framework aids in guiding experimental tuning of band topology in materials.

Abstract

Previous symmetry-based database searches have already revealed ubiquitous band topology in nature, while the destiny of band topology under symmetry-breaking is yet to be studied comprehensively. Here we first develop a framework allowing systematically ascertaining topological state evolution as expressed via a tree-like graph for magnetic/non-magnetic crystalline material belonging to any of the 1651 magnetic space groups. Interestingly, we find that specifying different ways of realizing symmetry-breaking leads to various contractions of the tree-like graph, as a new angle of comprehensively characterizing the correlation between a spontaneous symmetry-breaking and any symmetry-group-indicated physics consequence. We also perform a high-throughput investigation on the 1267 stoichiometric magnetic materials ever-experimentally synthesized to reveal a hierarchy of topological states along all continuous paths of symmetry-breaking (preserving the translation symmetry) from the parent magnetic space group to P1. The results in this work are expected to aid experimentalists in selecting feasible and appropriate means to tune band topology towards realistic applications and promote further studies on using tree-like graph to explore the interconnection between topology and other intriguing orderings.

Tables

Table 1. Table 1: The 236 magnetic topological materials predicted to be topological for at least 3 values of U 𝑈 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 𝑈 U . For example, ID = 1.308 in the last column uniquely corresponds to the magnetic topological insulator MnBi 2 Te 4 Bernevig  et al. ( 2022 ) .

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

Equations

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

Full text

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:

[TABLE]

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.

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.

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 [math] 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/.