Discovery of new topological insulators and semimetals using deep generative models

Introduction

Topological materials refer to materials with special topological arrangements in the geometric of electronic band structures, which can produce robust surface states and unconventional electromagnetic activities^1,2. The topology represents that the electronic structure of the materials will not change due to parameter tuning without opening the energy gap. These materials, encompassing topological insulators (TIs)^3,4,5 and topological semimetals (TSMs)^6,7,8,9, have captured considerable attention over recent decades due to their extraordinary properties. These materials showcase robust boundary states that are resilient to the perturbing effects of static disorder, distinguishing them from conventional materials. Moreover, their unique features, including the topological magneto-electric effect and anomalous transport phenomena such as the quantum anomalous Hall effect^10,11,12, highlight their potential to advance our understanding of condensed matter physics and spintronics. These novel electronic properties of topological materials have great potential in the development of dissipationless electronic¹¹ and spintronic devices¹³. Since the inception of this field, the quest to discover new TIs and TSMs has emerged as a pivotal and cutting-edge area of research. The advancements in this field have predominantly hinged on first-principles calculations anchored in topological band theory^14,15. In particular, the advent of theories like symmetry indictors¹⁶ and topological quantum chemistry (TQC)¹⁷ has led to the discovery of a large number of topological materials and the establishment of numerous databases^1,18,19 based on high-throughput computation. The burgeoning volume of data has made machine learning be used to solve the problems of theoretical classification frameworks in terms of computational speed²⁰, topology determination²¹, and low-symmetry structure classification²². Despite these notable advancements, current methodologies predominantly focus on identifying topological materials within pre-existing databases^{1,18,19,20,22}. There remains a significant challenge in the ability to discover novel topological materials that extend beyond the confines of pre-existing ones. This challenge is particularly pronounced in the realm of low-symmetry chiral Kramers-Weyl semimetals, where they will break the double spin entanglement guaranteed by the traditional symmetry rules such as Kramer’s theorem, thereby leading to the appearance of new topological phases^23,24. Nevertheless, materials with low-symmetry topological band structures are recognized as key candidates for achieving field-free switching and high energy efficiency technologies due to their unique electronic properties. The low-symmetry of these materials leads to unconventional spin polarization, which can be induced by current, enabling precise control of the magnetization state without the need for an external magnetic field^25,26,27. In light of these considerations, it becomes critically important to develop a robust, data-driven approach that can simultaneously generate and diagnose novel and stable TIs and TSMs which transcends the restrictions imposed by symmetry rules.

Inverse design^28,29,30,31 is a data-driven strategy with available data of desired material functionality, domain knowledge and artificial intelligence to discover materials that exhibit this functionality³¹. The data-driven strategy has emerged as a leading approach in the realm of novel materials research. A critical challenge in inverse design is searching for desired materials in countless possible materials with different properties which depend on chemical composition and crystal structure³². One such solution is the application of generative machine learning models in the design of new candidate structures. However, these generative models often grapple with the challenge of capturing translationally and rotationally invariant representations when dealing with crystalline materials^{28,33,34,35,36,37,38,39}. To overcome this limitation, the crystal diffusion variational autoencoder (CDVAE)⁴⁰ adopts diffusion models^41,42 to directly generate the atomic coordinates of crystalline lattice structures with these invariant representations and employs periodic graph neural networks (PGNNs)⁴³ as the backbone of variational autoencoders⁴⁴, enhancing its capacity to handle complex structures. These PGNNs enable CDVAE to capture the translational and rotational invariance of crystals by ensuring periodic rotation, reflection, and translation invariances, leading to higher stability and reduced screening costs. Compared to traditional methods such as trial and error, random search, and density functional theory (DFT) approaches, the advantage of diffusion-based models^45,46,47 is that they can efficiently generate diverse and highly realistic materials structure at an extremely low computational cost. This is because diffusion-based models generate novel materials by following the distribution of available materials in used datasets, which is usually called data-drawn discovery. This approach has been adeptly integrated into the inverse design framework, significantly enhancing the discovery process of an extensive array of innovative materials, including one-dimensional structures⁴⁸, two-dimensional (2D) materials³² and high-critical temperature superconducting materials⁴⁹. Despite these advancements, it is noteworthy that, within the current scope of our knowledge, the exploration and identification of new topological materials through the application of generative models represents a yet unexplored frontier.

In this work, we demonstrate that generative machine-learning models can also be used in the quest for new topological materials. We introduce a data-driven inverse design method CTMT tailored for the discovery of novel TIs and TSMs. CTMT synergistically combines several cutting-edge technologies: the aforementioned CDVAE, Topogivity²², interatomic potentials (IAPs) as realized in M3GNet⁵⁰, and TQC. This integrative approach enables CTMT not only to generate but also to effectively identify stable topological materials beyond pre-existing ones. Our method has proven its efficacy by finding 4 TIs and 16 TSMs that are absent in current material databases, including 4 chiral Kramers-Weyl semimetals. These findings exhibit the potential of CTMT as a universal tool for the exploration and discovery of novel topological materials, paving the way for uncovering even more diverse and rare material types.

Results

Workflow

Figure 1 illustrates the main framework of CTMT, including four sequential functional blocks: generation, filtering, stability verification, and topology type classification, for the systematic discovery and validation of new TIs and TSMs.

Discovery of new topological insulators and semimetals using deep generative models — **Fig. 1: The architecture and workflow diagram of CTMT with the number of passing candidates.**

Generation of Crystal Structures

The purpose of the first functional block is to generate new crystal structures. The used training dataset is from the topological materials database (https://www.topologicalquantumchemistry.fr/), as depicted in Fig. 1a, which includes 6109 TIs and 13,985 TSMs. The original database also includes 18,090 trivial materials, which were excluded from the training dataset. Notably, this same training dataset has been employed in recent studies^51,52,53. Supplementary Fig. S1 shows the atomic proportion percentage across the training dataset. The trained CDVAE model generates 10,000 highly realistic candidates of potential topological materials based on Langevin dynamic sampling⁴² (Fig. 1b, see Methods section for details). The generated 10,000 candidates are fed into the filter block, as shown in Fig. 1c.

Filtering Process

The filter block checks the novelty, legitimacy, and topologically nontrivial possibility of the candidates. The novelty check is performed by eliminating materials with the same chemical formula and structure in the dataset using the StructuresMatcher package of pymatgen⁵⁴. The main parameters for StructureMatcher were set as follows: lto at 0.2, stol at 0.3, and angle_tol at 5. The legitimacy of materials is determined by using the smart packages⁵⁵ to verify whether their stoichiometric chemical formulas satisfy charge neutrality and electronegativity balance. The third check uses pymatgen packages⁵⁴ to assess structural validity by examining if the bond length of these crystals is larger than 0.5 Å. After that, 4,715 valid candidates pass the first three checks and go through the ‘Topogivity’²² check, as shown in Fig. 1d. Topogivity is a machine-learned chemical rule for discovering topological materials²². A given material is diagnosed with high accuracy (typically > 80%) as topological nontrivial (trivial) if the weighted average of its element’s Topogivities is positive (negative). Our present work uses the criterion that the weighted average Topogivity should be greater than one to obtain more accurate topological nontrivial materials. In addition, candidates containing elements with 4f or 5f electrons are excluded due to the challenges in obtaining accurate results from DFT calculations, which arise from their complex electronic structures and the significant relativistic effects introduced by spin-orbit coupling (SOC). Furthermore, magnetic atoms are also excluded because of the additional computational procedures required to determine the magnetic ground state. The difficulty of generating materials with heavy elements and magnetic atoms is a potential obstacle that may influence results. This filtering gives 104 potential topologically nontrivial materials for further stability verification.

Stability verification

The stability verification block examines the stability of the candidates. First, DFT calculations are performed, as shown in Fig. 1e, to calculate the formation energy and energy above hull. If a candidate has a positive formation energy E_form ≥ 0 eV/atom or its energy above hull is E_hull ≥ 0.16 eV/atom, the candidate is thermodynamically unstable and will be removed. The 57 candidates with E_form < 0 eV/atom and E_hull < 0.16 eV/atom are potentially synthesizable (see Methods section for details). Since thermodynamic stability alone is not sufficient to ensure candidate stability, phonon spectrum calculations must be conducted to check whether there are imaginary phonon frequencies inside the 57 candidates. As the phonon spectrum calculation based on Vienna ab initial simulation package (VASP)⁵⁶ is a computationally intensive task, the pre-trained M3GNet⁵⁰ is directly integrated into CTMT without any further training to complete the verification. Compared to other machine-learning interaction potential methods such as MACE⁵⁷ and CHGNet⁵⁸, M3GNet can directly provide phonon spectrum, without converting predicted forces and energies into second-order force constants, while ensuring high prediction accuracy⁵⁹. Therefore, M3GNet is adopted in the present work for stability checks. As shown in Fig. 1f, the IAPs predicted by M3GNet is utilized to perform fast phonon spectrum calculations. This step filters out candidates with imaginary phonon frequencies and leaves 32 stable candidate materials (see Methods section for details).

Topology type classification

The fourth block, as shown in Fig. 1g, uses TQC on the Bilbao Crystallography Server¹ to ultimately determine the topology type of the materials, leading to the discovery of 20 new topological materials (Fig. 1h, see Methods section for details), including 16 TSMs and 4 TIs. TQC is a band theory of the structure of energy bands in crystals and links to the topological properties of crystals with electron orbitals at the Fermi level.

Topogivity

Figure 2 illustrates the distribution of atomic proportion percentages and Topogivity values across these 104 topologically nontrivial candidates after the Topogivity check, offering key insights into the diversity of their compositions. The trained CDVAE model generates candidates with a wide range of compositions, as evidenced by the fact that the atoms in these 104 materials span most elements with known Topogivity. Interestingly, elements such as oxygen (O), fluorine (F), phosphorus (P), chlorine (Cl), bromine (Br), and iodine (I) are absent in the 104 materials, because they have large negative values of Topogivity²². In the 32 stable candidates, 20 of them are confirmed to be topological, reaching an accuracy rate of 62.5%. Although this accuracy is lower than the classification accuracy of Topogivity (82.4%)²², CTMT discovers novel topological materials with extremely high fabrication feasibility. The integration of Topogivity into CTMT not only significantly reduces the material search space but also enhances the proportion of topological materials among the candidates recommended by the CDVAE model.

Stability

Figure 3a displays a detailed stability verification. The DFT calculations show that 77 crystals of the 104 candidates have formation energies E_form < 0 eV/atom and are thus considered thermodynamically stable, accounting for the percentage of 74%. In the 77 candidates, 57 materials have E_hull < 0.16 eV/atom with high synthesis possibility, which gives the percentage of 74% as well. Figure 3b, c show the energy distribution of E_form < 0 eV/atom for 77 materials and E_hull < 0.16 eV/atom for 57 materials, respectively. The formation energy is clustered around –0.4 eV/atom, indicating a general trend of energy stability. Similarly, the energy above the hull is distributed around 0 eV/atom. Notably, materials with E_hull = 0 eV/atom are of particular interest as they signify the ground state or the most stable configuration achievable. In this work, there are 11 materials with E_hull = 0 eV/atom as the phase diagrams shown in Supplementary Fig. S2, S3. Furthermore, the M3GNet integrated in CMTM estimates the phonon spectrums and gives only 32 structural stable and synthesis feasible candidates, out of the 57 potential candidates with a success rate of 56%, echoing with the prediction of 2D materials at 69%³².

Topological properties

The TQC methodology¹ is finally performed to ascertain the topological classification of each stable structure, resulting in 16 TSMs and 4 TIs. We use the TQC method to obtain the characters of all bands at all the relevant high-symmetry points (the maximal k-vectors). If the characters at the relevant high-symmetry points do not satisfy the compatibility conditions, it means they are enforced semimetals. Each identified TSM is classified as an enforced semimetal with Fermi degeneracy (ESFD). EFSD depends on whether they have a high-symmetry point degeneracy at the Fermi level. All 16 TSMs have a high-symmetry point degeneracy at the Fermi level, while the TIs exhibited distinct topological invariant numbers. The physical behaviors of bands in momentum space are interpreted using several kinds of topological invariant numbers. The topological invariant is a quantized number that characterizes the topological status of a given system, and the Chern number and Z₂ number are examples of the topological invariant numbers. Topological invariant numbers are the conserved quantities when any topological phase transitions do not occur⁶⁰. For the definition of TI in this context, from the theory of TQC¹⁷, if the set of bands below the Fermi level cannot be expressed as a linear combination of elementary band representations (EBRs), then it can be identified as a TI. It’s very important to note that sometimes the TI identified in this frame may lack a band gap, as we can also see many examples from the topological materials database^1,17,61. Unlike the conventional definition of an insulator, which requires a global band gap (i.e., a nonzero indirect band gap between the conduction band minimum and the valence band maximum). In TQC, it is sufficient for every high symmetry k point in the Brillouin zone to exhibit a direct band gap, such that the occupied and unoccupied states can adiabatically evolve into a global band gap¹⁷. TIs identified through TQC may exhibit band structures without a global gap, such as GeTa₃¹ and Bi²². The detailed characteristics of these topological materials are systematically cataloged in Table 1. Among them, the space group of the structure is determined using the SpacegroupAnalyzer method in the pymatgen⁵⁴ package. A particularly noteworthy finding is the identification of nonmagnetic chiral crystals, including 3 Kramers-Weyl semimetals with space group P1 and 1 Kramers-Weyl semimetal with space group C2. These Kramers-Weyl semimetals represent a new category of materials, hosting Kramers-Weyl fermions at time-reversal-invariant momenta^23,24. Kramers-Weyl fermions have attracted intense attention due to their unique physical properties including magneto-chiral dichroism⁶², large optical activity^63,64, quantized chiral charges⁶⁵ and negative longitudinal magnetoresistance⁶⁶ due to the intricate interplay of SOC, structural chirality and time-reversal symmetry. The previous research efforts aimed at discovering new topological materials always relied on symmetry rules^23,24, which met difficulties in dealing with low-symmetry as well as the chiral structures. However, our inverse design process, which does not rely on any symmetry-based rules, successfully identified a number of chiral structures with lower symmetry. This achievement underscores a significant advantage of our method, highlighting its potential to explore and uncover a broader spectrum of topological materials, particularly those with unconventional and complex structures.

Table 1 Formula, space group, Topogivity, topology classification and additional notes of 20 topological materials

Full size table

In Fig. 4, we highlight four novel topological materials with their crystal structures, band structures, and phonon spectra, which are considered the most likely to be synthesized. Among them, CdAu₅ (Fig. 4a, e, i) is a Kramers-Weyl semimetal with space group C2. In this material, the band splitting is observed at all points except at the time-reversal-invariant momenta. Li₂YBi₂ (Fig. 4b, f, j) and Zr₂ScC (Fig. 4c, g, k) are both TSMs with space group ({rm{P}}bar{3}{rm{m}}1) and R3m, respectively. Mg₄Pt₂ (Fig. 4d, h, l) is presented as a TI, characterized by a set of topological invariant numbers: Z_2w,1 = 1, Z_2w,2 = 1, Z_2w,3 = 1, Z₄ = 2, Z₂ = 0, Z₈ = 6. The absence of any obvious imaginary frequencies in their phonon spectra corroborates their structural stability⁶⁷. As an example of trivial band structures, Supplementary Fig. S4 presents the band structure and phonon spectra of crystal Ba₂Sn₄ generated by CTMT. This material, identified by TQC as a linear combination of EBRs (indicating trivial topology), is further confirmed to be stable through phonon analysis. The band structures of other topological materials identified in this study are available in Supplementary Fig. S5–S8, which provide detailed information on the structures and stability information of these materials.

**Fig. 4: Four highlighted topological materials.**

Discussion

Overall, by systematically combining CDVAE, Topogivity rule, M3GNet, and TQC, we have successfully developed a novel data-driven method CTMT for the inverse design of new TI and TSMs based on deep generative models. This innovative approach has led to the discovery of 20 novel and stable topological materials, including 16 TSMs and 4 TIs. Compared with the traditional methods^1,18,19 of directly determining the topology type of materials based on calculations, CTMT can reduce the calculation range to a smaller size and achieve higher accuracy while saving computational resources by preliminarily screening non-trivial materials based on Topogivity before calculation. In CTMT, the success rate of finding topological materials from stable materials is 62.5%, which is much higher than the traditional methods’ success rate of less than 30%. This outcome highlights the effectiveness of individual components within the CTMT framework in searching for new topological materials and proves the potential of CTMT in exploring all possible topological materials. Meanwhile, it is important to acknowledge that, due to the constraints related to computational accuracy and the stringent screening criteria applied for topology type and stability, there is a possibility that many potential topological materials within the generated dataset remain undiscovered. Our work opens up a novel and efficient path for finding groundbreaking topological materials, and holds great potential for the exploration of other advanced functional materials, such as topological superconductors, nodal line semimetals, and layered room temperature ferromagnetic materials. The future direction is to use topological properties as the generation condition in CTMT, so that it can consider both the stability of the crystal structure and the topological type of the material.

Methods

CDVAE training details

In this work, the dataset of topological materials was partitioned into training and validation subsets at a ratio of 8:2 for the CDVAE training. The backpropagation method was used in training with the Adam optimization algorithm and a learning rate set to 0.001. The training was completed after 800 epochs with the minimum loss on the validation set. During training, the hyperparameters are set to be consistent with those of the CDVAE training mp-20 dataset. The trained CDVAE model generates 10,000 candidates of potential topological materials, which are sent to a series of filters and thermodynamic stability checks by first-principles calculations based on DFT.

The CDVAE⁴⁰ model we used in this word is implemented by Xie et al. (https://github.com/txie-93/cdvae). This model incorporates DimNet++⁶⁸, adapted for periodicity as the encoder, and GemNet-dQ⁶⁹ as the decoder. Both the encoder and decoder are invariant to structure changes, comprising 2.2 million and 2.3 million parameters, respectively. The training dataset is extracted from The Topological Materials Database website (https://www.topologicalquantumchemistry.fr) by using the request package (https://requests.readthedocs.io) in Python to collect the crystal structure information and topology type. After that, the crystal structure information is converted into a crystallographic information file (CIF), and the Structure package in pymatgen⁵⁴ is used to feed the CIF into the CDVAE model. During training, we set the parameters of these networks with those used by Xie et al⁴⁰. Additionally, due to the absence of formation energy data and the unpartitioned test set in the collected topological material dataset, we adopted the hyperparameters from the modified MP-20 dataset. The modifications are specified as follows: We set the predicted property (“prop” in the code) to “scaled_lattice” and set the number of targets (“num_targets” in the code) to 6. Furthermore, we excluded the test dataset configuration and limited the maximum training epochs (“train_max_epochs” in the code) to 800. The sampling process was conducted by executing the file at https://github.com/txie-93/cdvae/blob/main/scripts/evaluate.py. Before execution, we configured the model path to the saved model parameter path and set the “tasks” parameter to “gen”, which generated 10,000 candidate structures.

For the analysis involving Topogivity and M3GNet, we employed their pre-trained models. The Topogivity values were directly retrieved from Fig. 2 of Ref. ²², and we included these values in Fig. 2 of our work as well. The IAPs used in M3GNet for phonon filtering were obtained by loading the pre-trained model from MP-2021.2.8-EFS (https://github.com/materialsvirtuallab/m3gnet/tree/main/pretrained/MP-2021.2.8-EFS). Following this, we applied the structure relaxation demo from the M3GNet package (https://github.com/materialsvirtuallab/m3gnet) to preform relaxation on the selected structures and calculated the phonon spectra using the phonopy package.

Calculation parameter settings

The VASP is used to carry out the DFT calculations with the exchange-correlation potential of the generalized gradient approximation in the Perdew-Burke-Ernzerh type. The convergence criteria of energy and force are 10^–6 eV and 0.01 eV/A, respectively and the cutoff energy for plane-wave expansion is 500 eV. The pre-trained M3GNet uses the 2×2 crystal superlattice cell and has the pre-sets of relaxation steps of 10,000 and a maximum force threshold of 0.0001 ({rm{eV}}/{text{AA }}). The phonon spectrum calculations employ the M3GNet force field and the phonopy packages⁷⁰. The “Check topological mat” module from TQC method⁵⁴ is integrated in CTMT and all calculations relevant to topological properties have been included with the SOC. The energy convergence precision is pre-set to 10^-8 eV. This rigorous approach facilitates the categorization of structures into TSM, TI, and linear combination of EBRs, and the latter indicates a topological trivial state with the set of bands below the Fermi level¹. The utilization of VASPKIT⁷¹ and the pymatgen packages⁵⁴ significantly expedites efficiently the processing of DFT data.