Introduction

With the deepening of theoretical and experimental research on topological materials, the development of topological semimetals (TSMs) has proved to be a prominent topic in condensed matter physics1,2,3,4,5, where the high-fold degenerate TSM was discovered as a new class of TSMs in recent years6,7. Comparing with traditional Dirac semimetals and Weyl semimetals, high-fold degenerate semimetals can host exotic massless fermionic excitations with higher fold of degeneracy. Theoretical predictions or experimental observations have identified the existence of three-8,9,10, four-11, six-12,13,14, and eightfold15,16 degenerate unconventional fermions. These new fermions are protected by crystal symmetry and can occur at the high symmetry k-points in the Brillouin zone (BZ). Their degeneracy is related to irreducible representations of the little group at the high symmetry k-points17,18. These newly discovered fermions not only contribute significantly to the advancement of fundamental science but also hold promising potential for the development of new materials with nontrivial properties, like topological catalyst with chiral structure19,20,21. Since the high-fold degenerate TSMs with chiral structures can host more surface states with extremely long Fermi arcs, more electrons with high mobility will lead to the excellent performance in catalysis. Thus, the exploration of high-fold degenerate TSMs with chiral structures will provide a favorable platform for topological catalysis22,23.

The chiral high-fold degenerate TSMs combine the advantages of both TSMs and chiral structure which lacks mirror planes, inversion center and rotation-reflection axes24,25. The high-fold degenerate fermions host large Chern number and long Fermi arcs across the entire surface BZ. The surface states connecting nontrivial fermions will have a large range of energy window that benefits from the absence of mirror symmetry. Moreover, the systems with high-fold degenerate fermions can also host Weyl fermions, which may exhibit special properties in chiral structures such as quantized circular photogalvanic effect (CPGE)26,27, gyrotropic magnetic effect28, etc. However, till now, a few chiral high-fold degenerate TSMs were discovered by theory or experiment, most of which come from space group P213 (No. 198). CoSi and PdBiSe family of materials are such TSMs with prominent topological properties and attract more attention of physicist in the field of topological materials12,29,30,31,32,33.

In this work, the high-throughput calculation based on density functional theory (DFT) were carried out to explore the topological high-fold degenerate points in chiral structures. The algorithm was first introduced in a manuscript, and 146 chiral structures were predicted to be high-fold degenerate semimetals or metals, which possess not only three-, four- or sixfold degenerate fermions in high symmetry k-points, but also a list of Weyl fermions locating at anywhere of the first BZ. Then, several representatives with different space groups were shown with topological properties, where Mn2Al3 with different kind of high-fold degenerate points was calculated thoroughly. The Weyl points and surface states in (001) surface of Mn2Al3 were observable, which confirm that the coexistence of high-fold degenerate fermions and conventional Weyl fermions in chiral high-fold degenerate TSMs, and make it a promising platform to investigate the novel physical properties of chiral fermions.

Results

Algorithm

The high-throughput calculation process is shown in Fig. 1. Starting with all the crystal structures from the material database Materials Project34, 18,506 materials belong to 65 chiral space groups are sifted. First, a series of screenings are conducted before DFT calculations. Since high-fold degenerate TSMs are metals or semimetals, the materials with large band gap are excluded in conformity with density of state. Considering the constraint of computation costs, the number of sites and elements in unit cell are limited to less than 40 and 4, respectively. In order to ensure the stability of chiral structures, according to the database Materials Project, the materials with energy above hull35,36,37 larger than 0.5 eV will be omitted here. Moreover, the systems with strongly correlated f-elections, namely lanthanides and actinides except element La, are also not considered because of the accuracy of DFT. After removing a few misleading structures containing data issues, 4368 materials are left totally.

Fig. 1: Workflow for the high-fold degenerate topological semimetals (TSMs) identification.
figure 1

The chiral crystal structures are loaded from the Material Project after removing the compounds containing strongly correlated f-electrons. Based on density functional theory, the nonmagnetic metals or semimetals are screened. By calculating the Chern number, we search the high-fold degenerate points around the Fermi level to confirm the topological metals or semimetals.

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After the screening processes above, we obtain a list of metals or semimetals with chiral structure that may host high-fold degenerate fermions. Among them, all the structures with the atomic magnetic moments exceeding 0.5 μB are regarded as magnetic compounds according to the self-consistent field calculation based on DFT. For the remaining 925 nonmagnetic metals with chiral structure, the band structure calculations considering spin–orbit coupling (SOC) with all high symmetry k-points are performed to explore the high-fold degenerate points near the Fermi level. Here we focus on the points that can exist within the energy range of [−0.5 eV, 0.5 eV] where the Fermi energy is set to 0 eV. For these points, we further construct the Wannier function automatically to calculate the monopole/anti-monopole charge of Berry curvature as a topological invariant C, which is expressed as an integral of Berry curvature Ω38 surrounding the degenerate point with extremely small radius:

$${{\mathcal{A}}}_{n}=i\left\langle n(k)\right\vert {\nabla }_{k}\left\vert n(k)\right\rangle$$
(1)
$$\Omega =\nabla \times {{\mathcal{A}}}_{n}$$
(2)
$$C=\frac{1}{2\pi }\int\Omega {d}^{2}k$$
(3)

where \({{\mathcal{A}}}_{n}\) is Berry connection for wave function \(\left\vert n(k)\right\rangle\). The effective algorithm to construct Wannier function automatically has been tested in our previous work39 successfully. Finally, only a total of 146 nonmagnetic chiral high-fold degenerate TSMs with high-fold degenerate points near the Fermi level are obtained.

High-fold degenerate TSM database

The dimension of the irreducible representation of the little group at high symmetry k-points corresponds to the degeneracy of high-fold degenerate points. So, we classify 146 high-fold degenerate TSMs by their space groups, and the high symmetry k-points which can host high-fold degenerate fermions are invariable for the materials with same space group. The details for each high-fold degenerate TSM candidate are given in Supplementary Table S1. We can find that all the chiral structures are belong to 14 space groups. Most of the materials can host fourfold degenerate points. However, none of them can possess eightfold degenerate points, which can only exist in 7 space groups (No. 130, 135, 218, 220, 222, 223, 230)6 that are not chiral. Moreover, the high-fold degenerate TSM CoSi and PdBiSe family of materials that have been verified are also listed in our database.

In the whole database, except CoSi and PdBiSe family of materials, we also discover a lot of new chiral structures with topological high-fold degenerate points near Fermi level. Here, we choose four materials with different space groups in Fig. 2 to investigate the distribution of topological high-fold degenerate fermions with nonzero monopole charge. Space group No. 197 can host fourfold degenerate points at the high symmetry k-points Γ and H whose point group is T, as shown in Fig. 2a for Ba2Ir3O9. Both of the monopole charge of fourfold degenerate points closest to Fermi level located at E = 0.168 eV and E = −0.065 eV are +4. For LiBi3O6 with space group No. 199 in Fig. 2b and La3SiBr3 with space group No. 214 in Fig. 2d, the fourfold and threefold degenerate points are observed at the same time. The fourfold and sixfold degenerate points can also coexist for space group No. 213, like Mg3Ru2 in Fig. 2c and Mn2Al3 in the following part. The high-fold degenerate points in other space groups in the database are listed in the supplementary. Moreover, the sum of monopole charge for entire BZ of nonmagnetic chiral crystal structure which have time reversal symmetry is zero, so most of these high-fold degenerate TSMs may have Weyl points.

Fig. 2: High-fold degenerate TSM representatives.
figure 2

Band structures of (a) Ba2Ir3O9 (space group No. 197) with two fourfold degenerate points, (b) Mg3Ru2 (space group No. 213) with two fourfold degenerate points and one threefold degenerate point, (c) LiBi3O6 (space group No. 199) with one fourfold degenerate point and one sixfold degenerate point, and d La3SiBr3 (space group No. 214) with one fourfold degenerate point and one threefold degenerate point.

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Discussion

In order to dissect the Weyl points in high-fold degenerate TSMs, we select Mn2Al3 with the space group P4132 (No. 213) as an example to analyze its topological properties thoroughly. Mn2Al3 has a cubic crystal structure with 20 atoms in one unit cell. It possesses 24 symmetry operations, including one threefold rotation axis along [111] direction and several screw axes. The chiral of the structure with space group P4132 is mainly embodied in the screw axes, as shown in Fig. 3a, b.

Fig. 3: Crystal and electronic structure of Mn2Al3.
figure 3

a Cubic lattice structure of Mn2Al3. The black dashed line represents the screw axis [3/4, 1/2, 0], while the red and blue solid lines with arrows show the chiral. b Top view of crystal structure in [111] direction. c Bulk Brillouin zone (BZ) and the projected surface BZ of (001) surface. d Band structure without spin–orbit coupling (SOC) which can host one fourfold degenerate point with C +2 at R. Black and magenta dashed circles are high-fold degenerate points and Weyl points, respectively. e Band structure with SOC which can host three fourfold degenerate points with C +2 or +4 at M, Γ and R. Two insets show the gap along the high symmetry k-path M − Γ and the sixfold degenerate point at R. The bands with two occupation numbers 92 and 94 are respectively labeled by red and blue.

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Based on the DFT calculations, we have investigated the band structures of Mn2Al3 firstly. In the absence of SOC, two kind of nontrivial points with nonzero monopole charge are observed, which is indicated in Fig. 3d. Because of the absence of inversion symmetry and mirror symmetry, the linear band crossings situating in Γ − X and Γ − M are two Weyl points with C = −1 and cannot be shaped into a nodal line structure. At the same time, a fourfold degenerate point with C = +2 is formed at the high symmetry k-point R and lies about 0.12 eV above Fermi level. After considering SOC effect, Fig. 3e reveals that the nontrivial points will be preserved and changed the location slightly. The Weyl point at Γ − X still locates at kx axis with a small shift. However, the Weyl point at Γ − M deviates a little from kx = ky direction and opens a band gap along the k-path. In addition, the fourfold degenerate point without SOC splits into one sixfold degenerate point located at E = 0.2 eV with C = +4 and one twofold degenerate point, which is protected by nonsymmophic symmetry. The twofold degenerate points at Γ and M without SOC will be doubled to become fourfold degenerate points located at E = 0.272 eV and 0.231 eV with C = +4 and +2, respectively. The degeneracies of these high-fold degenerate points located at the boundary of the three-dimensional (3D) BZ are protected by time reversal and nonsymmorphic symmetry, which is characterized by previous studies on Kramers–Weyl fermions17. At the same time, influenced by time reversal symmetry, the total monopole charge of the system must be zero. Thus, besides the nontrivial points along high symmetry k-path mentioned above, there must be other fermions to cancel the nonzero monopole charge.

We have screened the nontrivial points in entire BZ without and with SOC effect. The case without SOC has a total of four independent Weyl points W1−4,nosoc and one fourfold degenerated point H5,nosoc as shown in Table 1, some of which have been observed in band structure. The distribution for all nontrivial points in entire BZ is shown in Fig. 4a. The blue and green points represent the monopole charge with positive and negative sign, respectively. When SOC effect is considered, because the closed Weyl point for Fermi level on kx axis and the fourfold degenerate point at M point are produced by different bands and cannot be observed with one highest occupied band, we probe into the nontrivial fermions with two occupation number 92 and 94, which is labeled by red and blue band in Fig. 3e, respectively. Seven and 21 independent nontrivial fermions have been, respectively, found with the two occupied number, which is listed in detail in Table 2. For the band with the occupation number 92 near Fermi level, 6 independent Weyl points \({{\rm{W}}}_{1-6,soc}^{92}\) and one sixfold degenerate point \({{\rm{H}}}_{7,soc}^{92}\) at R are detected altogether, which is shown in Fig. 4d. At the same time, for the band with occupation number 94 located at about 0.2 eV, 18 independent Weyl points \({{\rm{W}}}_{1-18,soc}^{94}\), one sixfold degenerate point \({{\rm{H}}}_{19,soc}^{94}\) at R and two fourfold degenerate point \({{\rm{H}}}_{20-21,soc}^{94}\) at Γ and M are detected in all, which is shown in Fig. 4g. The other Weyl points which is not listed in two tables can be derived by the symmetry operations. Since the lack of mirror symmetry, the Weyl points with opposite chirality have different energy. Moreover, except for the Weyl points with C = ±1 and the high-fold degenerate points with larger C, we also find two independent quadratic Weyl points with C = −240.

Table 1 Weyl points and high-fold degenerate points of Mn2Al3 without spin–orbit coupling (SOC)
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Fig. 4: Weyl points of Mn2Al3.
figure 4

a–c Distribution of Weyl points, Berry curvature (BC) of kz = 0 plane and BC of kx = ky plane without SOC, respectively. d–f Distribution of Weyl points, BC of kz = 0 plane and BC of kx = ky plane with SOC at occupied number 92, respectively. g–i Distribution of Weyl points, BC of kz = 0 plane and BC of kx = ky plane with SOC at occupied number 94, respectively. kz = 0 and kx = ky plane are respectively shaped by brown and cyan plane in BZ. The blue (green) dots represent the Weyl points with positive (negative) chirality.

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Table 2 Weyl points and high-fold degenerate points of Mn2Al3 with SOC
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As is widely accepted, the integral of Berry curvature for a small sphere that wraps around Weyl point equal to the monopole charge carried by the Weyl fermion. The Berry curvature near Weyl fermions will exhibit as a source or a sink according to their chirality. From the distribution of Weyl points in Fig. 4a, d, g, we can find that all of the nontrivial points locate in kz = 0 plane or kx = ky plane, which is affected by the symmetry of space group P4132. The points at kx axis are forced by the fourfold rotation symmetry along [100] direction, and the points at [111] axis are forced by the threefold rotation symmetry. For other Weyl points can be attributed to the twofold rotation symmetry along [110] direction. Thus, for Mn2Al3, we calculated the Berry curvature of kz = 0 plane and kx = ky plane to observe the location of each Weyl point. Figure 4b, c, e, f, h, i shows the Berry curvature without SOC, with SOC when the occupation number is 92 and with SOC when the occupation number is 94, respectively. The Weyl points with negative chirality perform as the divergence of Berry curvature, while the positive ones perform as the convergence of Berry curvature. For example, the Berry curvature in kz = 0 plane exhibits two independent Weyl points W1−2,nosoc in Fig. 4b, and the two points change into \({{\rm{W}}}_{1-2,soc}^{92}\) with a slight difference in position and quantity after considering SOC effect in Fig. 4e, which also can be discovered in the band structure.

Furthermore, the Fermi arc surface states connecting projected nontrivial fermions are one of the unique properties of TSM. From the (001) surface BZ projected shown in Fig. 3c, we can find that the sixfold degenerate point at R and the fourfold degenerate point at M are projected into one high symmetry k-point \(\bar{{\rm{M}}}\), and the fourfold degenerate point at Γ still locates at \(\bar{\Gamma }\) point. Unfortunately, these three high-fold degenerate points are all merged into the bulk states, but we can observe the surface states cased by Weyl points explicitly, which is shown in Fig. 5a. From the dispersion of surface state, we notice one projected Type-I Weyl point W1 derived from \({{\rm{W}}}_{1,soc}^{92}\) at \({{\rm{E}}}_{{{\rm{W}}}_{1,soc}^{92}}\)= 0.182 eV lies on the high symmetry k-point \(\bar{\Gamma }\), and two projected Type-I Weyl points W2 derived from \({{\rm{W}}}_{2,soc}^{92}\) at \({{\rm{E}}}_{{{\rm{W}}}_{2,soc}^{92}}\)= 0.064 eV locate along the high symmetry k-path \(\bar{{\rm{X}}}\)–\(\bar{\Gamma }\). The surface states with energy fixed at the energy of W1 and W2 can also discover the Fermi arcs connecting Weyl points obviously, as indicated in Fig. 5b, c. For the system with chiral structures, the Weyl points with opposite chirality have different energy because of the lack of mirror symmetry. So, only part of the Fermi arc connecting one projected Weyl point can be observed in one energy cut. Each projected Weyl point is formed by two equivalent Weyl points in 3D material, so there are 4 and 2 Fermi arcs emerge from W1 projected by \({{\rm{W}}}_{1,soc}^{92}\) with C = −2 and W2 projected by \({{\rm{W}}}_{2,soc}^{92}\) with C = −1, respectively. Taking the rotation symmetry into account, there are 4 Fermi arcs for W1 in the first BZ totally, and 16 Fermi arcs connecting W2 satisfied the symmetry in itself. Although having just part of surface states including one Weyl point, the Fermi arcs still extend about half of the BZ which is sufficient for experimental measurement.

Fig. 5: Surface states of Mn2Al3.
figure 5

a Surface energy dispersion for (001) surface. The green filled dots represent the projected Weyl points with negative chirality, and the blue hollow dots represent the projected high-fold degenerate points with positive C. b, c (001) surface Fermi arcs with energy fixed at projected Weyl points W1 and W2. There are 4 and 2 Fermi arcs emerge from W1 projected by \({{\rm{W}}}_{1,soc}^{92}\) with C = − 2 and W2 projected by \({{\rm{W}}}_{2,soc}^{92}\) with C = −1, respectively.

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In summary, through high-throughput calculations base on DFT, we have established the database about nonmagnetic high-fold degenerate TSMs with chiral structure. Most of them may contain both high-fold degenerate points and Weyl points to make sure that the sum of monopole charge for the first BZ is zero, such as Mn2Al3. Furthermore, the surface states connecting nontrivial fermions can have natural long surface Fermi arcs, which can be recognized in experiment conveniently. Beside the database of TSM, the high-fold degenerate fermions with chiral structure may provide the platform to study topological catalysis, and the coexisting Weyl points are also favorable for the research in topological properties like quantized CPGE.

Methods

Computational details

To investigate the electronic structures and properties, first-principles calculations in this work were carried out based on DFT using the package Vienna Ab initio Simulation Package (VASP)41. The generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof functional (PBE)42 was chosen as exchange-correlation potential. The Monkhorst–Pack k-mesh of the BZ in the self-consistent process was set to be 8 × 8 × 8. The lattice constants were fully relaxed until the force for each atom was less than 10−3 eV/Å. In order to investigate the topological properties, a tight-binding Hamiltonian based on the maximally localized Wannier functions (MLWF)43 were generated by the most orbits near the Fermi level. The surface states were explored with the Green’s function method44,45 using a half-infinite boundary model. The topological invariants and Berry curvature were calculated by the WannierTools package46.