Research Article

Beyond Dirac and Weyl fermions: Unconventional quasiparticles in conventional crystals

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Science  05 Aug 2016:
Vol. 353, Issue 6299, aaf5037
DOI: 10.1126/science.aaf5037

Structured Abstract

INTRODUCTION

Condensed-matter systems have recently become a fertile ground for the discovery of fermionic particles and phenomena predicted in high-energy physics; examples include Majorana fermions, as well as Dirac and Weyl semimetals. However, fermions in condensed-matter systems are not constrained by Poincare symmetry. Instead, they must only respect the crystal symmetry of one of the 230 space groups. Hence, there is the potential to find and classify free fermionic excitations in solid-state systems that have no high-energy counterparts.

RATIONALE

The guiding principle of our classification is to find irreducible representations of the little group of lattice symmetries at high-symmetry points in the Brillouin zone (BZ) for each of the 230 space groups (SGs), the dimension of which corresponds to the number of bands that meet at the high-symmetry point. Because we are interested in systems with spin-orbit coupling, we considered only the double-valued representations, where a 2π rotation gives a minus sign. Furthermore, we considered systems with time-reversal symmetry that squares to –1. For each unconventional representation, we computed the low-energy k · p Hamiltonian near the band crossings by writing down all terms allowed by the crystal symmetry. This allows us to further differentiate the band crossings by the degeneracy along lines and planes that emanate from the high-symmetry point, and also to compute topological invariants. For point degeneracies, we computed the monopole charge of the band-crossing; for line nodes, we computed the Berry phase of loops encircling the nodes.

RESULTS

We found that three space groups exhibit symmetry-protected three-band crossings. In two cases, this results in a threefold degenerate point node, whereas the third case results in a line node away from the high-symmetry point. These crossings are required to have a nonzero Chern number and hence display surface Fermi arcs. However, upon applying a magnetic field, they have an unusual Landau level structure, which distinguishes them from single and double Weyl points. Under the action of spatial symmetries, these fermions transform as spin-1 particles, as a consequence of the interplay between nonsymmorphic space group symmetries and spin. Additionally, we found that six space groups can host sixfold degeneracies. Two of these consist of two threefold degeneracies with opposite chirality, forced to be degenerate by the combination of time reversal and inversion symmetry, and can be described as “sixfold Dirac points.” The other four are distinct. Furthermore, seven space groups can host eightfold degeneracies. In two cases, the eightfold degeneracies are required; all bands come in groups of eight that cross at a particular point in the BZ. These two cases also exhibit fourfold degenerate line nodes, from which other semimetals can be derived: By adding strain or a magnetic field, these line nodes split into Weyl, Dirac, or line node semimetals. For all the three-, six- and eight-band crossings, nonsymmorphic symmetries play a crucial role in protecting the band crossing.

Last, we found that seven space groups may host fourfold degenerate “spin-3/2” fermions at high symmetry points. Like their spin-1 counterparts, these quasiparticles host Fermi surfaces with nonzero Chern number. Unlike the other cases we considered, however, these fermions can be stabilized by both symmorphic and nonsymmorphic symmetries. Three space groups that host these excitations also host unconventional fermions at other points in the BZ.

We propose nearly 40 candidate materials that realize each type of fermion near the Fermi level, as verified with ab initio calculations. Seventeen of these have been previously synthesized in single-crystal form, whereas others have been reported in powder form.

CONCLUSION

We have analyzed all types of fermions that can occur in spin-orbit coupled crystals with time-reversal symmetry and explored their topological properties. We found that there are several distinct types of such unconventional excitations, which are differentiated by their degeneracies at and along high-symmetry points, lines, and surfaces. We found natural generalizations of Weyl points: three- and four-band crossings described by a simple k · S Hamiltonian, where Si is the set of spin generators in either the spin-1 or spin-3/2 representations. These points carry a Chern number and, consequently, can exhibit Fermi arc surface states. We also found excitations with six- and eightfold degeneracies. These higher-band crossings create a tunable platform to realize topological semimetals by applying an external magnetic field or strain to the fourfold degenerate line nodes. Last, we propose realizations for each species of fermion in known materials, many of which are known to exist in single-crystal form.

Fermi arcs from a threefold degeneracy.

Shown is the surface density of states as a function of momentum for a crystal in SG 214 with bulk threefold degeneracies that project to (0.25, 0.25) and (–0.25, –0.25). Two Fermi arcs emanate from these points, indicating that their monopole charge is 2. The arcs then merge with the surface projection of bulk states near the origin.

Abstract

In quantum field theory, we learn that fermions come in three varieties: Majorana, Weyl, and Dirac. Here, we show that in solid-state systems this classification is incomplete, and we find several additional types of crystal symmetry–protected free fermionic excitations. We exhaustively classify linear and quadratic three-, six-, and eight-band crossings stabilized by space group symmetries in solid-state systems with spin-orbit coupling and time-reversal symmetry. Several distinct types of fermions arise, differentiated by their degeneracies at and along high-symmetry points, lines, and surfaces. Some notable consequences of these fermions are the presence of Fermi arcs in non-Weyl systems and the existence of Dirac lines. Ab initio calculations identify a number of materials that realize these exotic fermions close to the Fermi level.

Condensed-matter systems have recently become a fertile ground for the discovery of fermionic particles and phenomena predicted in high-energy physics. Starting with graphene and its Dirac fermions (1), continuing to Majorana fermions in superconducting heterostructures (27), and most recently, with the discovery of Weyl (816) and Dirac (1722) semimetals, solid-state physics has proven to abound in analogs of relativistic free fermions. There is, however, a fundamental difference between electrons in a solid and those at high energy: For relativistic fermions, the constraints imposed by Poincaré symmetry greatly limit the types of particles that may occur. The situation in condensed-matter physics is less constrained; only certain subgroups of Poincaré symmetry—the 230 space groups (SGs) that exist in three-dimensional (3D) lattices—need be respected. There is the potential, then, to find free fermionic excitations in solid-state systems that have no high-energy analogs.

Here, we theoretically identify and classify these exotic fermions, propose experiments to demonstrate their topological character, and point out a large number of different classes of candidate materials in which these fermions appear close to the Fermi level. We consider materials with time-reversal (TR) symmetry and spin-orbit coupling. We found that three of the SGs host half-integer angular momentum fermionic excitations with threefold degeneracies, stabilized by nonsymmorphic symmetries. The existence of threefold (and higher) degeneracies has long been known from a band theory perspective (2327). Our purpose is to elucidate their topological nature. Here, we show that all of the threefold degeneracies either carry a Chern number ±2 or sit at the critical point separating the two Chern numbers. In two closely related SGs, the combination of TR and inversion results in sixfold degeneracies that consist of a threefold degeneracy and its time-reversed partner, a “sixfold Dirac point.” There are two other types of sixfold degeneracies, both of which are distinct from free spin-5/2 particles. We also discuss eightfold degeneracies, which were recently introduced in (28). Here, we prove that there exists a finer classification of the eightfold degenerate fermions. Last, we show that for the threefold, as well as for a class of fourfold degeneracies, the low-energy Hamiltonian is of the form k · S, where S is the vector of spin-1 or -3/2 matrices. This provides a natural generalization of the recently discovered Weyl fermion Hamiltonian, k · σ (26, 29, 43, 44, 53, 60, 63, 66, 67).

We enumerate all possible three-, six-, and eightfold degenerate fermions in the subsequent sections. We include a symmetry analysis for each degeneracy type; an exhaustive search of the 230 SGs (23) guarantees that the list is complete. For each type of fermion, we investigated the low-energy k · p Hamiltonian allowed by the SG symmetries (23, 2931). This determines which degeneracies carry nontrivial Berry curvature or host exotic symmetry-enforced degeneracies along high-symmetry lines or planes in the Brillouin zone (BZ) (3234). We explore the role these features play in transport and surface properties. Last, using ab initio techniques, we predict existing material realizations for each distinct type of fermion, where they appear close to the Fermi level.

SGs with three-, six-, and eight-band crossings

The guiding principle of our classification is to find irreducible representations (irreps) of the (little) group of lattice symmetries at high-symmetry points in the BZ for each of the 230 SGs; the dimension of these representations corresponds to the number of bands that meet at the high-symmetry point and is one of the characteristics of the fermion type. Because we are interested in fermions with spin-orbit coupling, we consider only the double-valued representations; TR symmetry is an antiunitary that squares to –1. The results of our search are summarized in Table 1. All the SGs include nonsymmorphic generators, and all representations are projective; these are in fact necessary ingredients for the three-, six-, and eightfold degenerate irreps (35).

Table 1 Summary of the fermion types identified in this paper in solid-state systems.

La indicates the type of lattice (cP, cubic primitive; cB, cubic body-centered; and tP, tetragonal primitive), d indicates the maximum degeneracy at the relevant k point in the presence of TR symmetry. Group generators are defined in (35).

View this table:

We found that SGs 199, 214, and 220 may host a 3D representation at the P point in the BZ [the high-symmetry points are defined in (35)]. These SGs have a body-centered cubic Bravais lattice, and the P point is a TR noninvariant point at a corner of the BZ (that is, P ≠ –P). All three of these systems host a complementary threefold degeneracy at –P because of TR symmetry; Kramers’s theorem requires this to be the case. SG 214 is distinct in that the threefold degeneracy at –P persists even if TR symmetry is broken, because the P and –P points are related by a twofold screw rotation in the full-symmetry group.

In the presence of TR symmetry, six SGs can host sixfold degeneracies. In all cases, these arise as threefold degeneracies that are doubled by the presence of TR symmetry. Four of these—SGs 198, 205, 212, and 213—correspond to simple-cubic Bravais lattice, and the sixfold degeneracy occurs at the TR invariant R point at the corner of the BZ. The other two sixfold degeneracies occur in SGs 206 and 230 at the P point. Although this point is not TR invariant, these SGs are inversion symmetric, and hence all degeneracies are doubled.

Last, in agreement with previous work (28), we found that seven SGs may host eightfold degeneracies. However, as shown below, the resulting fermions fall into distinct classes. Two of these, SGs 130 and 135, have a tetragonal Bravais lattice; these are special in that they require eightfold degeneracies at the TR invariant A point. In addition, SGs 222, 223, and 230 may host eightfold degeneracies. SGs 222 and 223 are simple-cubic, and an eightfold fermion can occur at the R point in the BZ; for SG 230, it occurs at the TR invariant H point.

There are two more SGs that can host eightfold degeneracies, SG 218 and SG 220. These differ from the others in that they lack inversion symmetry. Energy bands away from high-symmetry points need no longer come in pairs. SG 218 has a simple cubic Bravais lattice, and an eightfold degeneracy may occur at the R point. In SG 220, the degeneracy may occur at the H point.

Low-energy effective models

For each of the band crossings in Table 1, we computed a low-energy expansion of the most general Hamiltonian consistent with the symmetries of the little group near the degeneracy point, k0, in terms of δkkk0. Full details of the constructions are in (35). Representative plots of the band dispersion along high-symmetry lines are shown in Figs. 1 to 3, where higher-order terms have been included for the sake of clarity.

Fig. 1 Energy dispersion near a threefold degeneracy at the P point.

(A and B) Shown are threefold degenerate points in (A) SGs 199 and 214 and (B) SG 220. In the latter case, pairs of bands remain degenerate in energy along the high-symmetry lines Embedded Image.

Fig. 2 Energy dispersion near a sixfold degeneracy.

(A to C) Six-band crossings in (A) SG 205; (B) SG 206 and 230; and (C) SGs 198, 212, and 213. In SGs 198, 212, and 213, bands become degenerate in pairs along the faces δki = 0 of the BZ. In SGs 205, 206, and 230, all bands are twofold degenerate owing to inversion symmetry.

Fig. 3 Energy dispersion near an eightfold degeneracy.

(A to C) Eightfold degenerate points in (A) SGs 130 and 135; (B) SGs 222, 223, and 230; and (C) SGs 218 and 220. (A) and (B) show pairwise degeneracy owing to inversion symmetry. In addition, in (A), two degenerate bands form fourfold degenerate line nodes along the edges of the BZ. In (C), the eightfold degeneracy splits into four nondegenerate and two doubly degenerate pairs of bands along the high-symmetry Embedded Image lines.

We began by analyzing the threefold degeneracy points. The linearized k · p Hamiltonian for SGs 199 and 214 takes the formEmbedded Image (1)where Embedded Image is a real parameter; without loss of generality, we set the zero of energy at zero throughout and omit an overall energy scale. The bands are nondegenerate away from δk = 0, unless Embedded Image for integer n, in which case bands become degenerate along the lines Embedded Image. Although the locations of these degeneracies in Embedded Image change in the presence of higher-order terms, they identify two topologically distinct phases. First, for Embedded Image, the δk ≠ 0 Hamiltonian is adiabatically connected to the one with Embedded Image for sufficiently small Embedded Image. At this value of φ, the Hamiltonian takes the formEmbedded Image (2)where the matrices Si are the generators of the rotation group SO(3) in the spin-1 representation. This shows that our threefold fermion is a fermionic spin-1 generalization of an ordinary Weyl fermion. The translation phases of the nonsymmorphic little-group symmetries have effectively converted a half-integer spin representation into an integer-spin representation. The three bands ψ±, ψ0 of the Hamiltonian have energies Embedded Image. Furthermore, the Chern numbers of each of these bands evaluated over any closed surface enclosing the degeneracy point are Embedded Image and v0 = 0. These Berry fluxes characterize the entire phase Embedded Image.

At Embedded Image, the v = 0 band becomes degenerate, with both ψ± bands at different points in momentum space; these degeneracies transport Berry curvature between ψ+ and ψ. The formation of line nodes at the transition is an artifact of linearization: When higher-order terms are included in the Hamiltonian, the line nodes break up into sets of four single Weyl nodes, which carry Berry curvature away from the degeneracy point. The properties of all the phases for the other values of Embedded Image can be derived from those for Embedded Image; all regions feature bands with Chern number ±2 (35). Thus, this three-band crossing has the topological character of a double Weyl point (36), but the dispersion of a single Weyl point; this behavior is facilitated by the trivial (v = 0) band passing through the gapless point (Fig. 1A, energy spectrum).

Having identified a fermion with a spin-Embedded Image k · S Hamiltonian, it is natural to ask whether there exist similar particles for higher values of angular momentum j. Our fermion classification rules out the possibility of j ≥ 2 because these would either have degeneracy greater than eight, which we have ruled out via an exhaustive search, or would have appeared on our list. In (35), we present a full classification of j = 3/2 fermions, which we found can be stabilized by either symmorphic or nonsymmorphic symmetries; we found seven SGs that can host this excitation. Three of these overlap with groups mentioned earlier: SGs 212 and 213 can host spin-3/2 fermions at the Γ point, and SG 214 can host a spin-3/2 fermion at the Γ and H points.

The threefold degeneracy in SG 220 is distinct from that in SGs 199 and 214. The linear-order k · p Hamiltonian readsEmbedded Image (3)This threefold degeneracy sits at a critical point in the phase diagram for SG 199 described in the previous paragraph. Consequently, pairs of two bands are degenerate along the lines Embedded Image (Fig. 1B). Mirror and threefold rotation symmetry dictate that—unlike for SGs 199 and 214 discussed above—these line nodes persist to Embedded Image orders in the k · p expansion (35). The line nodes are characterized by the holonomy of the wavefunction (Berry phase), around any loop encircling the line, given by w = –1.

Next, we consider the sixfold band degeneracies. We start with SGs 205, 206, and 230, in which TR symmetry Embedded Image times inversion I forces all bands to be twofold degenerate (Fig. 2, A and B). In SGs 206 and 230, the k · p Hamiltonian can be written asEmbedded Image (4)Thanks to Embedded Image, there is no abelian Berry curvature (Chern number) associated with these degeneracies. Instead, we can consider the non-Abelian SU(2)-valued holonomy [Wilson loop (37, 38)] of the wavefunction in each twofold degenerate pair of bands. Evaluated along C2 symmetric loops, the eigenvalues of the SU(2) holonomy matrix wind twice—and in opposite directions—around the unit circle as a function of position in the BZ (35). As gauge-invariant quantities, these eigenvalues are in-principle measurable (39, 40), hence their winding provides a meaninful topological classification.

Unlike the previous cases, SG 205 contains inversion symmetry in the little group of the R point. This forces the effective Hamiltonian to be quadratic in δk. However, it is still related to H199 byEmbedded Imagewhere Fk) is a diagonal matrix whose entries are Embedded Image and all cyclic permutations of δki. Because of its quadratic coordinate dependence, H205k) has only bands of zero net Berry flux, and Wilson loop eigenvalues do not wind.

We conclude our analysis of the three- and sixfold fermions with SGs 198, 212, and 213. Unlike the other six-band systems, these lack inversion symmetry, and so host six bands with distinct energies. The linearized k · p Hamiltonians may be written asEmbedded Image (5)where δk′ = (δkz, δkx, –δky), and b is an arbitrary parameter. The six eigenstates of these Hamiltonians have distinct energies except along the faces of the BZ, where the spectrum degenerates into pairs related by the composition of a nonsymmorphic C2 rotation and time reversal; this degeneracy is shown in Fig. 2C. Because this symmetry is antiunitary and squares to –1, these degeneracies are stable to higher-order terms in k · p.

Next, we examine the eightfold fermions. In SGs 130 and 135, Embedded Image symmetry mandates doubly degenerate bands. Close to the A point, the linearized k · p Hamiltonian readsH130 = H135 = δkz(aσ2σ3σ3 + bσ2σ3σ2 + cσ2σ3σ1) + δkx(–dσ1σ3σ0 + eσ1σ2σ3 + fσ1σ2σ2 + gσ1σ2σ1) + δky(–dσ3σ3σ0 + eσ3σ2σ3 + fσ3σ2σ2 + gσ3σ2σ1) (6)where a, b, ... g are real-valued parameters. This Hamiltonian has fourfold degenerate line nodes along lines δki = δkj = 0 with Embedded Image, which follow the BZ edges (Fig. 3A). This is seen by noting that the matrices multiplying any given δki are part of a Clifford algebra. These lines are generally protected by composites of time reversal and nonsymmorphic mirror symmetry. Thanks to Embedded Image symmetry, abelian Berry phase of these line nodes vanishes. However, they can be characterized by the two (–1) eigenvalues of the SU(2) Wilson loop encircling them.

Similarly, for SGs 222, 223, and 230, we haveEmbedded Image (7)and a similar expression holds for H230 after a permutation of every δk. Besides the Embedded Image double degeneracy of all bands, there are no additional degeneracies (Fig. 3B).

Last, we examine the eightfold degeneracy in SGs 218 and 220. Because both of these cases lack Embedded Image, they have eight nondegenerate bands away from the high-symmetry point. However, there is a degeneracy along high-symmetry lines emanating from it. Along lines Embedded Image, the eightfold degeneracy splits into four singly degenerate bands and two pairs of doubly degenerate bands. In addition, along lines where two of the δki are zero, and along lines where δki = δkj, δkk = 0, there are four pairs of doubly degenerate bands. Unlike SGs 198, 212, and 213 above, however, there are no additional degeneracies along high-symmetry planes. The spectrum is shown in Fig. 3C. The k · p Hamiltonian is given in (35).

Experimental signatures

We now consider how to experimentally detect the topological character of the new fermions. We start with the threefold degeneracy in SGs 199 and 214. Because the degeneracy at the P point carries net Berry flux Embedded Image, the surface spectrum will host two Fermi arcs that emerge from the surface projection of the P point (13), similar to those that appear from double Weyl points (36). In the presence of TR, an additional threefold degeneracy exists at the –P point at the same energy; its surface projection will be the origin for two more Fermi arcs. Furthermore, because the monopole charge is invariant under the action of time reversal, materials in this symmetry group will always exhibit Fermi arcs. Whether these arcs are masked by other spurious Fermi pockets depends on the details of the band structure; the arcs will nevertheless always exist. These four Fermi arcs must terminate on the surface projection of four Weyl points (or two double Weyl points), which must exist elsewhere in the BZ. These can be identified with the Weyl points that drive the topological phase transition between Chern number v = ±2: At the phase transition, two Weyl fermions emerge from the threefold degeneracy to carry away the Fermi arcs of the v = +2 phase, whereas two other Weyl points emerge as the endpoints of the Fermi arcs for the v = –2 phase. We have verified this with a toy tight-binding model for SG 214. In the surface density of states for a surface in the Embedded Image direction in the first surface BZ (Fig. 4), a pair of Fermi arcs is visible, emanating from the surface projections of the P point. Breaking TR symmetry with an external Zeeman field will split each threefold degeneracy into a number of Weyl points (35); generically, these will be a mix of type I and type II Weyl fermions (41). In SG 199, this will destroy the exact degeneracy between the P and –P points, but the degeneracy will persist in SG 214 for perturbations invariant under Embedded Image.

Fig. 4 Tight-binding surface states for SG 214.

Shown is the surface density of states for a surface in the Embedded Image direction, calculated by using the open-source PythTB package (70). The x axis gives the angle around a circle δk(θ) = 0.05[g2cos(θ) + g3sin(θ)] surrounding the surface projection of the P point, and the y axis is energy. Here, g2 = 2π(1, 0, 1) and g3 = 2π(1, 1, 0) are the surface reciprocal lattice vectors. Two chiral Fermi arcs can be clearly seen traversing the bulk gap. (Inset) The atoms in nine unit cells with lines to indicate the nonzero hopping amplitudes. Each unit cell consists of four atoms with three p orbitals per atom. Only p orbitals with intersite spin-orbit coupling are included.

In addition to Fermi arc surface states, the threefold fermions will exhibit anomalous negative magnetoresistance and a chiral anomaly distinct from that of either a single or double Weyl point. For weak magnetic fields, semiclassical considerations (42) suggest that the magnetoresistance in SGs 199 and 214 match that of a double Weyl point (36), although the density of states corresponds to a linear dispersion. At large magnetic fields, the Landau level spectrum for the threefold fermions displays two chiral modes; unlike the case of an ordinary (single or double) Weyl point, the spectral flow of these chiral modes does not pass through zero momentum but instead flows to kz → ±∞, assisted by the presence of the nearly flat trivial band. We show the numerically computed Landau level spectrum in Fig. 5 and the analytical derivation of the spectrum at the exactly solvable Embedded Image point in (35).

Fig. 5 Landau level spectrum of the threefold degenerate fermion.

The spectrum is shown as a function of δkz with magnetic field along the z direction. The two chiral modes are distinguished by being monotonic until they reach an avoided crossing (inset) with the family of bands near zero energy. Because of the avoided crossing, both chiral modes have a spectral flow that goes through kz → ±∞.

Because the rest of the fermion types we identified here do not host net Berry curvature, there is no guarantee of topologically protected surface states in the strictest sense. However, the nontrivial Berry phase associated with the line node in SG 220 implies the presence of a “Fermi drum ” surface state (33, 43), although this will not be robust to breaking of the crystal symmetry in the bulk. Also, despite the name, these states need not be flat—or even nearly flat—in energy. Similarly, the non-abelian Berry phase associated to the Dirac lines in SG 130 and 135 suggests the existence of pairs of drumhead states. And in the presence of an external magnetic field (or strain perturbation), these Dirac lines may be split to yield any of the usual gapless topological phases: Weyl, Dirac, and line node semimetals. This opens up the possibility of tuning topological Dirac semimetals in these materials with a Zeeman field, similar to the recent progress made with field-created Weyl semimetals in half-Heusler materials (8, 12). Last, all of the fermion types are also detectable via angle-resolved photoemission spectroscopy (ARPES) [for example, similar experimental results in (44)] and through quantum oscillation experiments.

Material realizations

We propose candidate materials (45, 46) that realize each of the types of fermions identified here near the Fermi level. In (35), we provide many more examples that require doping to bring the Fermi level to the band crossing but that, in the cases in which the fermions are below the Fermi level, are still observable in ARPES experiments. We have computed the band structure of each candidate (Figs. 6 to 9) to confirm that the desired band crossings exist and are relatively close to the Fermi level. We performed electronic structure calculations within density-functional theory (DFT) as implemented in the Vienna ab initio simulation package (47) and used the core-electron projector-augmented-wave basis in the generalized-gradient method (48). Spin-orbital coupling (SOC) is accounted for self-consistently. Unless otherwise noted, the materials we propose to host these fermions have been synthesized as single crystals.

Fig. 6 Materials exhibiting threefold fermions near the Fermi level.

(A and B) The band structures of (A) Ag3Se2Au (SG 214), where the threefold band crossing is 0.5 eV below the Fermi level, and (B) Pd3Bi2S2 (SG 199), where the threefold crossing is almost exactly at the Fermi level.

Fig. 7 Compounds in SG 220 display three- and eightfold fermions near the Fermi level.

(A and B) The band structures of (A) Ba4Bi3 and (B) La4Bi3, both in SG 220. Both materials exhibit both a threefold fermion at the P point and an eightfold fermion at the H point. The threefold crossings split into a doubly degenerate line node and a single band along the cuts of the BZ shown in the figure; hence, the band crossings appear to be only twofold degenerate. Similarly, the eightfold crossing splits into four doubly degenerate line nodes and hence appears as a fourfold crossing. The band crossings are within 0.25 eV of the Fermi level.

Fig. 8 Sixfold fermions at the R point.

(A to F) The band structures of the representative compounds (A) MgPt, (B) AsPdS, and (C) K3BiTe3 in SG 198 and (D) Li2Pd3B (SG 212), (E) Mg3Ru2 (SG 213), and (F) PdSb2 (SG 205). Because SG 205 contains inversion symmetry, all bands are doubly degenerate.

Fig. 9 Eightfold fermions at the A and R points.

Eightfold fermions are visible at the A point in (A) CuBi2O4 (SG 130), (B) PdBi2O4 (SG 130), and (C) PdS (SG 135) and at the R point in (D) CsSn (SG 218), (E) Ta3Sb (SG 223), (F) LaPd3S4 (SG 223), and (G) Nb3Bi (SG 223). Because (A) to (C) and (D) to (G) have inversion symmetry (in addition to TR symmetry), all bands are doubly degenerate. In addition, in (A) to (C), the fourfold degenerate line nodes are visible along the A-M line, as described in the main text. [(D), inset] The eight-band crossing splits into four doubly degenerate lines along R-X and into four nondegenerate and two pairs of double degenerate bands along Γ-R.

We begin with an exotic three-band fermion in SG 199, in the material Pd3Bi2S2, which exists in single-crystal form (49). The band crossing at the P point is only 0.1 eV above the Fermi level, and its position could be further tuned with doping (Fig. 6B).

Next, we consider the exotic three-band fermion in SG 214 in Ag3Se2Au, which can be grown as a single crystal (50). As shown in Fig. 6A, although the spin-1 Weyl point is located 0.5 eV below the Fermi level, there are fourfold degeneracies at the Γ and H points located only 0.02 eV below the Fermi level, and there are no other bands in the vicinity; this remarkable material exhibits a k · S type spin-3/2 Hamiltonian close to the Γ and H points.

SGs 220 and 230 can host these types of fermions at both the P and H points. In SG 220, we found threefold and eightfold fermions in Ba4Bi3 (51) and La4Bi3 (Fig. 7, A and B) (52) . In the latter case, the band crossings are <0.1 eV from the Fermi level. These materials are parts of the families of compounds A4Pn3 and R4Pn3 [A, Ca, Sr, Ba, and Eu; R, rare-earth element (La, Ce); Pn (pnictogen), As, Sb, and Bi], which are also potential candidates.

MgPt (53) is a near ideal example of a six-band fermion in SG 198; As shown in Fig. 8A, the band crossing is ~0.3 eV above the Fermi level and isolated from other bands. More examples can be found in the families of PdAsS (54) and K3BiTe3 (Fig. 8, B and C) (55). These band crossings are ~0.7 eV below and 0.5 eV above the Fermi level, respectively. Similar fermions can be found closer to the Fermi level in the compounds Li2Pd3B (SG 212) (56) and Mg3Ru2 (57), shown in Fig. 8, D and E.

The quadratic six-band fermions in SG 205 can be found in PdSb2 (58), as shown in Fig. 8F, as well as in the similar compounds FeS2 and PtP2.

The eight-band fermions required to exist in SG 130 sit almost exactly at the Fermi level in CuBi2O4 and are isolated from all other bands (Fig. 9A). This material has a filling of 180 = 8*22 + 4 electrons per unit cell, and hence it could be a realizable example of a filling enforced semimetal (59) above its Neèl temperature, if not for interaction effects, which appear to make the material insulating (60). Other bismuth oxides in this SG may be more promising candidates for eight-fold fermions; PdBi2O4 (61), which exists in single-crystal form, is shown in Fig. 9B, and two predicted compounds are shown in (35)

Eight-band fermions are also required to exist in SG 135. One example is shown in Fig. 9C in PdS, sitting 0.25 eV above the Fermi level. This material is naturally insulating but could be potentially doped. It has been observed in polycrystalline form (62).

The eight-band fermions predicted to occur in SG 218 should exist in CsSn (63) and CsSi (64) and, more generally, in the class AB for A = K, Rb, Cs and B = Si, Ge, Sn; the band structure of CsSn shows its distinct splitting into four twofold degenerate bands in the kx = kz direction away from the R point in Fig. 9D. There is a similar eight-band fermion at the H point in SG 220, which is shown in Fig. 7 for Ba4Bi3 (51) and La4Bi3 (65).

The eight-band fermions predicted to occur in SG 223 are exhibited in the candidates X3Y, where X is either Nb or Ta and Y is any group A-IV or A-V element in the β-tungsten structure A15, as well as in the family MPd3S4, where M is any rare-earth metal. The band structures for Ta3Sb (powder) (66) and LaPd3S4 (67) show the eight-band crossing within nearly 0.1 eV of Fermi level (Fig. 9, E and F). Nb3Bi (powder) (68), which has two eightfold fermions within 0.1 eV of the Fermi level, is shown in Fig. 9G. It is possible that other materials that host these types of fermions near the Fermi level may be identified through an exhaustive database search of filling-enforced semimetals.

Outlook

Here, we have analyzed all possible exotic fermion types that can occur in spin-orbit coupled crystals with TR symmetry going beyond the Majorana-Weyl-Dirac classification. By virtue of their band topology, these fermions can play host to interesting surface states, magnetotransport properties, and ARPES signatures. Growth of many of the material candidates mentioned above—including AsPdS, La3PbI3, La4Bi3, LaPd3S4, and Ta3Sb—may be possible and, if successful, should yield fruitful results in ARPES and magnetotransport experiments.

As we have emphasized throughout, nonsymmorphic crystal symmetries were essential for stabilizing these fermions; it is the presence of half-lattice translations that allow spin-1/2 electrons to transform under integer spin representations of the rotation group, yielding threefold and sixfold degeneracies. The types of fermions we identified here also provide an explicit realization of the nonsymmorphic insulating filling bounds derived recently in (59, 69), as follows. Thanks to the presence of TR symmetry, we know that our threefold fermions must occur in connection with an additional nondegenerate band, so that all bands have Kramers partners at TR invariant momenta; this suggests that the minimal band connectivity in these SGs is four, which is consistent with the filling bounds. Similarly, the sixfold fermions arise from doubling a threefold degeneracy, each of which comes along with an aforementioned additional nondegenerate band. We thus expect—and indeed confirm—that the minimal insulating filling is eight in these cases. Last, as noted in (28), the eightfold degenerate fermions saturate the filling bound for those SGs.

Looking ahead, there are several open questions that deserve future attention. First, gapping these degeneracies by breaking the symmetries that protect them can lead to distinct symmetry-protected topological phases, with new classes of 2D gapless surface modes. Furthermore, our symmetry analysis can be extended to crystals with magnetic order, and hence with interactions. This requires an investigation of representations of the 1191 remaining magnetic SGs.

Materials and methods

For all 230 space groups, we identified unconventional excitations by identifying the allowed little-group representations at each high-symmetry point in the Brillouin zone. Next, we used k · p perturbation theory to derive the most general symmetry-allowed low-energy Hamiltonian for each fermion type; we computed topological invariants from eigenvectors obtained by diagonalizing these Hamiltonians. We did our ab initio electronic structure within DFT as implemented in the Vienna ab initio simulation package (37) and used the core-electron projector-augmented-wave basis in the generalized-gradient method (49). SOC is accounted for self-consistently.

Supplementary Materials

www.sciencemag.org/content/353/6299/aaf5037/suppl/DC1

Supplementary text

Figs. S1 to S9

Tables S1 to S3

References (7194)

References and Notes

  1. Additional analyses are available as supplementary materials on Science Online.
Acknowledgments: The authors thank A. Alexandradinata, T. Neupert, A. Soluyanov, and A. Yazdani for helpful discussions. M.G.V. acknowledges the Fellow Gipuzkoa Program through Fondo Europeo de Desarrollo Regional Una Manera de hacer Europa and the FIS2013-48286-C2-1-P national project of the Spanish Ministerio de Economía y Competitividad. B.A.B. acknowledges the support of the Army Research Office (ARO) Multidisciplinary University Research Initiative (MURI) on topological insulators, grant W911-NF-12-1-0461, ONR-N00014-11-1-0635, NSF CAREER DMR-0952428, NSF–Materials Research Science and Engineering Center (MRSEC) DMR-1005438, the Packard Foundation. and a Keck grant. R.J.C. similarly acknowledges the support of the ARO MURI and the NSF MRSEC grants. The authors declare no competing interest.
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