Multifold nodal points in magnetic materials

The prediction and observation of Weyl^1–8 and Dirac^9–13 fermions catalyzed an intense theoretical and experimental search for topological nodal semimetals in condensed matter systems.^14–21 These systems provide a solid state realization of chiral^8,22 and gravitational anomalies^23,24 and exhibit many novel physical properties, such as gapless Fermi arc surface states,¹ extremely large magnetoresistance,²⁵ and giant nonlinear optical response.^26,27

Topological semimetals also illustrate the interplay between symmetry and topology: Weyl fermions can exist without any symmetry but are forbidden by the combination of time reversal and inversion symmetry. On the other hand, Dirac fermions rely on a combination of crystal symmetries for their existence and their dispersion depends on the symmetries that protect them.²⁸ Crystal symmetries can also protect nodal lines.^29–39 

In Ref. 40, we completed the catalog of nodal point fermions that exist in systems with time-reversal symmetry and spin-orbit coupling by showing that nonsymmorphic symmetries can stabilize three-, six-, and eightfold degeneracies at the corners of the Brillouin zone and, furthermore, that chiral fourfold degeneracies are possible. (We use chiral to refer to a nodal point that is a source/sink of the Berry curvature. This should not be confused with the usage to refer to symmetry groups with only orientation-preserving operations,^41,42 which we refer to as structural chirality.) The multifold fermions include higher-spin analogs of Weyl and Dirac fermions, which cannot be realized in high-energy physics because they necessarily violate the Poincaré symmetry. Recently, multifold fermions have been observed in several experimental systems.^43–46 

However, the search for nodal semimetals has mostly focused on nonmagnetic materials, where time reversal symmetry is present. There are two practical reasons for this: first, hundreds of thousands of known compounds are searchable by their crystal structure but not by their magnetic order, in the International Crystal Structure Database (ICSD).⁴⁷ Second, the phase space of nonmagnetic symmetries is much smaller: there are 230 nonmagnetic space groups but 1421 additional magnetic space groups. Yet, increasingly, attention is being shifted to magnetic compounds. Several magnetic materials have been predicted to host Weyl^48–57 and other multifold fermions.^58–60 Symmetry indicators and filling constraints have also been computed for the magnetic space groups.⁶¹ In addition, magnetic materials are desirable for their large degree of tunability: by varying temperature⁶⁰ or applied magnetic field,^62,63 potentially many different phases can be realized.

Recent experimental progress combined with the tunability of magnetic materials motivates the study of nodal fermions in magnetic systems. In this work, we classify the degeneracy of fermionic quasiparticles that can occur in the magnetic space groups with spin-orbit coupling: as in the nonmagnetic groups, we find that three-, six-, and eightfold degeneracies can occur, in addition to the two- and fourfold degeneracies already known. We identify which of these are chiral and thus can be detected by their gapless surface states and optical response.⁴² We then describe candidate materials to realize these fermions.

As part of our investigation, we uncover space groups whose symmetries protect eightfold Rarita-Schwinger (RS) fermions. While “Rarita-Schwinger Weyl” fermions—the chiral halves of RS fermions—have already been predicted to exist in condensed matter systems,^40,64,65 the full RS fermion had not been explicitly identified. We find that these fermions can exist in both magnetic and nonmagnetic space groups (the latter case was included in our classification in Ref. 40 but not identified as such). Since the RS fermion is nonchiral, it can, but need not, display surface Fermi arcs (the same is true for Dirac fermions^66,67).

Our work is applicable to materials with a commensurate magnetic order that breaks time-reversal symmetry but preserves the product of time reversal and a spatial symmetry. We consider the limit where the magnetic order is “frozen” such that the system is described by weakly interacting single-particle excitations. We leave the treatment of incommensurate magnetic ordering and dynamical magnetic moments to future work.

The paper is outlined as follows: in Sec. II, we define the magnetic space groups. In Sec. III, we derive the magnetic space groups that can host 3-, 6-, and 8-dimensional fermions. In Sec. IV, we describe the Rarita-Schwinger fermion. In Secs. V and VI, we present the k · p Hamiltonians that describe the threefold and sixfold degenerate fermions. We present candidate material realizations in Sec. VII. A discussion and outlook is in Sec. VIII.

Of the 1421 magnetic space groups mentioned in Sec. I, 230 of them correspond to the ordinary space groups with no antiunitary symmetry operations. Since these “Type I” groups do not contain time-reversal symmetry, they can describe crystals with commensurate magnetic order. However, they describe only a limited set of magnetic orderings. A more complete set is given by space groups that lack time reversal symmetry but do contain other antiunitary elements. For a thorough introduction to the magnetic space groups, we refer the reader to Refs. 68–70. Here, we give a brief overview of the different types of space groups, with and without antiunitary symmetries, following the notation of Ref. 68.

Given a Type I group, H, adding time-reversal symmetry, $T$⁠, as a generator yields a Type II (nonmagnetic) group, $H∪TH$⁠. Thus, there are also 230 Type II groups.

The remainder of the magnetic space groups contain an antiunitary generator that is the product of time reversal and a unitary symmetry operation but do not possess time-reversal symmetry itself. These groups can be written in the form

where H is a Type I group and g₀ ∉ H is a unitary element. If the set g₀H does not (does) contain a fractional lattice translation, the group is referred to as a Type III (Type IV) group. H is referred to as a halving subgroup of G. Examples of each type are shown in Fig. 1.

If we add time reversal as a generator to G, we obtain the Type II group,

where G′ is the Type I group defined by

G″ describes the symmetry of G above the Néel temperature. The relationship between H, G, G′, and G″ is depicted pictorially in Fig. 2.

G and H contain the same lattice translations and hence have the same unit cell (even in the Type IV case, where g₀H contains a fractional lattice translation, this translation by itself is not an element of G; it is only an element of G when followed by $T$⁠). In the Type III groups, G′ and G″ have the same unit cell as G and H, while in the Type IV groups, G′ and G″ have a different (smaller) unit cell than G and H.

We present examples of G, G′, G″, and H in a Type III and Type IV group.

Let G be the Type III group depicted in Fig. 1. G is generated by the product of a sixfold rotation and time-reversal symmetry as well as the translation vectors of the honeycomb lattice; the sixfold rotation, which is not itself a symmetry of G, plays the role of g₀ in Eq. (1). The unitary subgroup, H, of G, is generated by a threefold rotation and translations. In contrast, the unitary group G′ is generated by the sixfold rotation and translations (and therefore contains elements that are not in G, as depicted in the Venn diagram in Fig. 2). The Type II supergroup G″ has the same generators as G′ and, in addition, time-reversal symmetry; therefore, G″ contains G, G′, and H as subgroups.

As a second example, let G be the Type IV group depicted in Fig. 1. G is generated by a fourfold rotation about the origin, translations by $x^±y^$⁠, and the product of time reversal and a translation by $x^$ [the translation by $x^$ plays the role of g₀ in Eq. (1) and is not a symmetry of G since it moves an up spin to a down spin in Fig. 1]. The unitary subgroup, H, of G, is generated by a fourfold rotation and translations by $x^±y^$⁠. The unitary group G′ is generated by the generators of H and, in addition, the translation by $x^$⁠. Hence, G′ has a smaller unit cell than G, as is always the case in a Type IV group. The Type II supergroup G″ has the same generators as G′, in addition to time-reversal symmetry.

There are two conventions to describe Type III and Type IV groups, the Belov-Nerenova-Smirnova⁷¹ (BNS) and the Opechowski-Guccione⁷² (OG) notations. We utilize the BNS notation because it more naturally accommodates the magnetic Brillouin zone, which is crucial for correctly determining degeneracies in the band structure of magnetic compounds. While we again refer the reader to Refs. 68–70 for details, we briefly summarize the BNS notation here. Each magnetic space group is assigned both a symbolic label (e.g., $P43′32′$ or P_I2₁3) and a numeric label (e.g., 212.61 or 198.11). The symbolic label uses the Hermann-Mauguin notation for space groups, with two additions: (1) a prime after an operation indicates that the operation is not in the space group, but the product of that operation and time-reversal symmetry is, and (2) in the Type IV groups, the lattice has a subscript indicating the magnetic Bravais lattice. The numeric label consists of two numbers: in the Type III groups, the number before the decimal point corresponds to the unitary group G′, while in Type IV groups, the number before the decimal point corresponds to the unitary subgroup H. The number after the decimal point is an index.

We are interested in degeneracies that occur at high-symmetry points in the Brillouin zone. An n-fold degeneracy at a point k occurs if there is an n-dimensional irreducible corepresentation of the little group, G_k, which consists of all symmetry operations that leave k invariant modulo a reciprocal lattice vector (an equivalence denoted by “≡”),

where equality in momentum space is defined modulo a reciprocal lattice vector and translations act trivially in momentum space. Corepresentations are the generalization of representations to groups with antiunitary elements; for simplicity, we will refer to irreducible corepresentations as irreps, whether or not they contain antiunitary elements.

In the Type I groups, G_k is always unitary and its irreps are easily looked up on the Bilbao Crystallographic Server (BCS)⁷³ or in Ref. 68. Previously,⁴⁰ we used these tables to find the high-dimensional irreps for the Type I and II groups; the results are reproduced in Table I. However, the analogous searchable tables do not yet exist for magnetic groups. In this section, we explain how to efficiently find the irreducible corepresentations of dimension 3, 6, and 8 in the Type III and IV groups from the tables in Ref. 70, which list the dimensionality of all irreps at all high-symmetry points in the remaining 1191 magnetic space groups. We need only consider the high-symmetry points because their little groups contain the little groups of all other points in the BZ. We focus exclusively on the “double-valued” irreps, where 2π rotations are represented by $−I$ (⁠$I$ denotes the identity matrix); these describe systems with non-negligible spin-orbit coupling, which we expect to apply to many magnetic systems.

In the Type III groups, G, H, G′, and G″ share the same BZ. Thus, if there is a three-, six-, or eightfold degeneracy at k in G, then there must also be a three-, six-, or eightfold degeneracy at k in the Type II group G″, which has more symmetry than G. The Type II groups with three-, six-, or eightfold degeneracies were enumerated by us in Ref. 40 and reproduced in Table I for convenience. Consequently, we can find all magnetic Type III space groups with these degeneracies by scanning the tables in Ref. 70 for Type III groups whose numeric label starts with one of the G″ listed in Table I (recall that G″ and G′ have the same numeric label and that the tables in Ref. 70 are sorted in BNS notation, which lists Type III groups by G′). The results are shown in Table II.

We need to be more clever for the Type IV groups because their BNS label contains H but not G′. In addition, since G′ and G have different Brillouin zones, it is not immediately evident how to find the irreps of G_k from those of G′_k.

We start by relating irreps of G_k to those of H_k. As discussed in Sec. II, G and H always have the same BZ. Hence, H_k is exactly the unitary part of G_k for every k. Thus, if G_k is unitary, then G_k = H_k. It follows that the irreps of G_k (and their dimensionality) are exactly the irreps of H_k, which can be looked up on the BCS.

Now, suppose G_k contains antiunitary elements. Then, there must exist an h₀ ∈ H such that

where g₀ is defined in Eq. (1); otherwise, G_k would be unitary. We prove in  Appendix A that G_k takes the form

It follows that the dimensionality of the irreps of G_k is determined by examining each irrep of H_k and determining whether its dimension stays the same or doubles when an antiunitary generator (⁠$Tg0h0$⁠) is added to the group. The algorithm for determining this is detailed in Refs. 68 and 70. However, because implementing the algorithm directly is time consuming, we now explain how to use the existing tables⁷⁰ to efficiently search the Type IV magnetic groups for a group, G, and high-symmetry point, k, such that G_k has a 3-, 6-, or 8-dimensional irrep.

Since each irrep of G_k either has the same or double the dimensionality of an irrep of H_k, a necessary condition for G_k to have a 3-, 6-, or 8-dimensional irrep is that H_k have a 3- or 4-dimensional irrep (double-valued irreps of H_k are at most 4-dimensional^40,68,73). We have enumerated H_k with 3- or 4-dimensional double-valued irreps in Tables VI and VII in  Appendix B. For reference, we have also enumerated the 3-, 4-, and 6-dimensional single-valued irreps in  Appendix C. These can be used to extend our results to magnetic materials with negligible spin-orbit coupling.

The tables in Ref. 70 list the irreps of the magnetic space groups at all high-symmetry points in the BNS notation, which, for Type IV groups, is sorted by the unitary subgroup H. Thus, for each H and k in Tables VI or VII, we search Ref. 70 for the magnetic groups derived from H and determine which of these have the 3-, 6-, or 8-dimensional double-valued irreps we seek. The results are in Table II.

The cases where G_k has a 3-, 6-, or 8-dimensional irrep or corepresentation are listed in Table II.

In the following sections, we describe the results in more detail. In particular, in Sec. IV, we prove which of the 8-dimensional irreps describe Rarita-Schwinger fermions. In Sec. V, we prove that all of the 3-dimensional irreps carry a charge-2 monopole of the Berry curvature, which we named a “spin-1 Weyl” in Ref. 40; consequently, they are accompanied by surface Fermi arcs. (All magnetic threefold degeneracies are nondegenerate away from the nodal point, in contrast to some nonmagnetic cases.) In Sec. VI, we show that all of the 6-fold fermions are nonchiral and topologically equivalent to two opposite-chirality copies of the spin-1 Weyl; we appropriately refer to them as “spin-1 Dirac” fermions. In general, we find that the Type IV groups have an extra symmetry constraint that pins the k · p Hamiltonian to an exactly solvable point, which would not be achievable in the nonmagnetic space groups without fine-tuning.

A generalization of Dirac fermions, Rarita-Schwinger (RS) fermions, are relativistic particles transforming in the spin-3/2 representation of the rotation group.⁷⁴ Although never observed as fundamental particles, RS fermions play an essential role in supergravity: the supersymmetric partner to the graviton is predicted to be a RS particle.⁷⁵ The Hamiltonian of a massless RS fermion takes the little-known form⁷⁶ 

where J is the vector of spin-3/2 matrices and τ_z is a Pauli matrix acting in the chirality basis. Note that since the spin-3/2 matrices do not all anticommute, the second term in the parentheses is nonanalytic in k. Because of this, we should not expect to reproduce this term in a k · p expansion for a crystal Hamiltonian. It has recently become fashionable^42,64,65 to refer to quasiparticles described by the Hamiltonian

which corresponds to two decoupled spin-3/2 fermions of opposite chirality, as RS fermions; we shall refer to these as RS^* fermions to emphasize the differences. While the Hamiltonians H_RS and H_RS^* share the same eigenstates, they are not topologically equivalent. To see this, let us focus on the + chirality. Using the results of Refs. 40 and 77, we see that the positive energy bands in the RS Hamiltonian are degenerate (as mandated by Lorentz invariance), with Chern numbers +3 and +1 (we define the Chern number of a band to be the Chern number of a putative Fermi surface that resides in that band). In the RS^* Hamiltonian, on the other hand, all bands are nondegenerate away from k = 0. Though distinct, both of these are homotopic to different phases of the “spin-3/2 fermion” introduced in Ref. 40. In Ref. 40, the RS spectrum was realized as the critical point in the phase diagram of the spin-3/2 fermion, while the RS^* Hamiltonian was realized at a fine-tuned point.

We will now search for quasiparticles near band degeneracies in crystals that can reproduce the dynamics of the 8 × 8 RS and RS^* Hamiltonians. Let us first revisit the k · p Hamiltonians for the eightfold degeneracies in the nonmagnetic Type II space groups $Pn3¯n1′$ (222) and $Pm3¯n1′$ (223). While we will primarily reproduce the results of Ref. 40, we will do so in a way that generalizes to a treatment of multifold degeneracies in other (magnetic) space groups and sheds new light on some of the complicated k · p Hamiltonians presented there.

To begin, we note that the two Type II space groups $Pn3¯n1′$ and $Pm3¯n1′$ (the 1′ indicates that time-reversal symmetry is a generator) can be expressed in terms of Type II halving subgroups,

where “≈” denotes a group isomorphism and we have used the standard notation⁶⁸ where {R|t} denotes a point group operation R followed by a translation t; I indicates the inversion symmetry operation. In Table III, we list the high symmetry points in these space groups with three-, six-, four-, or eightfold degeneracies and their degeneracy with and without spin-orbit coupling. Let us focus first on the group $Pn3¯n1′$ (SG 222); the halving subgroup in this case is, from Eq. (9), the symmorphic space group P4321′ (SG 207), with point group O. Because this group is symmorphic, we can determine its little group representations at any k point by examining the commutation relations between the generators of the point group O = ⟨C_4z, C_3,111⟩, $T$⁠, and the coset representative $g=I|121212$⁠. In particular,

For any point k that is not a time reversal invariant momenta (TRIM), the element g will not be in the little group G_k. Thus, for these k points, the little groups in space groups P4321′ and $Pn3¯n1′$ coincide. For the TRIM points, however, the size of the little group in SG $Pn3¯n1′$ is doubled relative to SG P4321′. Equations (11)–(13) show that at a TRIM point, k, a representation, Δ_k, of the little group in SG $Pn3¯n1′$ satisfies the following commutation relations:

where $K$ is the complex conjugation operation and {t_i} are a basis for the cubic Bravais lattice. Since we are primarily interested in the eightfold degenerate fermion in this space group, let us focus on the R point. To prove that the low-energy theory near R is described by the RS or RS^* Hamiltonian, we will build the eight-dimensional representation Δ_k out of the four-dimensional spin-3/2 representation of the R point in P4321′. We will show how the inclusion of g in the little group requires the degeneracy of the spin-3/2 representation to double and that the k · p Hamiltonian takes a RS form.

The reduced coordinates of the R point are $(12,12,12)$⁠. The little group at R, which we denote by $GR222$ (the superscript 222 indicates the numeric symbol for $Pn3¯n1′$⁠), contains the entirety of space group P4321′ as well as the coset representative g. By taking $k=(12,12,12)$ in Eqs. (14)–(19), the commutation relations show that at the R point, the representative of g must commute with the representative of C_3,111 and anticommute with the representatives of C_4x, C_4y, and C_4z. Let us focus on the spin-3/2 representation, ρ, of the little group $GR207$ (the superscript 207 indicates the numerical symbol for P4321′) and use it to build a representation Δ of $GR222$⁠. We will show first that we cannot consistently define a four-dimensional representation matrix for g and, hence, conclude that the fourfold degeneracy at R in space group P4321′ doubles when going to space group $Pn3¯n1′$⁠. For convenience, we introduce the spin-3/2 matrices,

We also introduce two-by-two Pauli matrices σ_i and τ_i such that τ_i exchange the two-by-two blocks in the spin-3/2 basis and σ_i act within the blocks. Using these matrices, the spin-3/2 representation ρ can be written as

Since g² = E and since ρ(g) must anticommute with ρ(C_4z), it must be that

Since ρ(g) must also commute with $ρ(T)$⁠, it must be that α = 0, and so our putative ρ(g) takes the form

However, upon attempting to enforce the constraint

we find δ = γ = 0. Thus, we conclude that there does not exist a four-dimensional representation of $GR222$ in space group $Pn3¯n1′$ that reduces to the spin-3/2 representation in space group P4321′. Therefore, upon adding the symmetry g to space group P4321′, the four dimensional representation at R must double in size. One convenient choice of basis for this eight-dimensional irreducible representation Δ in space group $Pn3¯n1′$ is given by

where we have introduced a new set of Pauli matrices μ that act to exchange the different copies of ρ in this eight-dimensional irrep. It is clear that by breaking inversion symmetry and reducing $GR222$ to $GR207$⁠, the representation Δ restricts to

We have thus constructed an eight-dimensional representation at the R point in space group $Pn3¯n1′$ in a basis that makes explicit the importance of the inversion symmetry g. We will now impose these symmetry constraints on the k · p Hamiltonian expanded about R. We will see that our choice of basis makes the connection to the RS and RS^* Hamiltonian manifest.

To begin, let us write the k · p Hamiltonian near the R point as

in terms of the 4 × 4 block matrices A, B, and C. Next, From Eq. (33), we deduce that

where H₂₀₇(k) is the spin-3/2 generalized k · S Hamiltonian presented in Ref. 40. It can be in either the RS or RS^* phase. The next simplest symmetry to impose is the composite $gT$⁠, which yields the constraint

which implies that

Equation (37) fully determines C(k): we see that if B(k) = 0, then the Hamiltonian H_R(k) would be homotopic to the nonchiral RS and RS^* Hamiltonians. However, Eq. (38), combined with the constraints of C₃ symmetry, also allow for a single nontrivial solution,

Thus, we see that the general k · p Hamiltonian for the eightfold degeneracy at the R point of space group $Pn3¯n1′$ consists of two spin-3/2 fermions of opposite chirality, coupled with the k-dependent matrix B(k). In the language of high-energy physics, this corresponds to a nonchiral RS or RS^* fermion with a k-dependent mass B(k)μ_y. The matrix μ_z is the chirality matrix, which, due to the presence of the mass term, is not a conserved quantity, unless c = 0. When c is not zero, the chirality of a given state is a time-dependent quantity that oscillates with momentum-dependent frequency ω_C = 2c|k|.

The essential ingredients for the preceding analysis in this section to hold are a cubic point group and the combined antiunitary symmetry $gT$⁠. Consequently, if g and $T$ are broken by magnetism, but their product is preserved, the eightfold degeneracy at R need not split. Instead, the form of the coupling B(k) will change.

To fully enumerate the RS fermions that can occur in crystals, we now examine the eightfold degeneracies in other magnetic and nonmagnetic space groups. We begin with the nonmagnetic groups.

We showed in Ref. 40 (concurrently with Ref. 78) that there are seven nonmagnetic Type II space groups with eightfold degeneracies in the presence of SOC; these are listed in Table IV. We now show that all of the eightfold fermions in the cubic groups are of the Rarita-Schwinger type, while those in the tetragonal space groups are distinct.

First, the little group of space group $Pm3¯n1′$ at the R point is isomorphic to $GR222$⁠. As such the analysis and k · p Hamiltonian derived in Secs. IV A and IV B also apply to the RS fermions in this space group. Next, recall that the Type II space groups $I4¯3d1′$ (220) and $P4¯3n1′$ (218) also host eightfold degeneracies, at the H and R points, respectively; in Ref. 40, these were found to have a quite complicated k · p Hamiltonian. However, they yield the coset decompositions

which show that in both cases the eightfold fermions emerge from an antichiral doubling of the four dimensional irreducible corepresentation $1F¯2F¯$ of the structurally chiral tetrahedral group T (isomorphic to the little cogroup of P231′ at the R point and I2₁3′ at the H point). As this corepresentation is the restriction of the spin-3/2 representation of the structurally chiral octahedral group O, we can follow the same logic as in Sec. IV B to find the k · p Hamiltonian

Here, H₁₉₅(k) is the k · p Hamiltonian for the spin-3/2 fermions in tetrahedral space groups, first given in a different basis in Refs. 42 and 43. It is a generalization of the Hamiltonian H₂₀₇(k), with one additional free parameter, due to the fact that SG 195 lacks {C_4z|000} as a generator. We find that the chiral coupling B(k) is a three-parameter generalization of Eq. (39), whose explicit form is not particularly enlightening. Because the k · p Hamiltonian H₂₁₈(k) reduces to H_R(k) of Eq. (36) for a particular choice of parameters, we conclude that space groups $P4¯3n1′$ and $I4¯3d1′$ can also host RS fermions.

Finally, the space group $Ia3¯d1′$ also hosts an eightfold degeneracy. This space group is obtained by adding inversion symmetry to the space group $I4¯3d1′$⁠. This restricts the k · p Hamiltonian in $Ia3¯d1′$ to the form of Eq. (36). Hence, this space group hosts RS fermions as well.

Thus, we have shown that all eightfold fermions in the cubic, nonmagnetic space groups are of Rarita-Schwinger type. We now prove that the eightfold fermions in the tetragonal space groups P4/ncc1′ (130) and P4₂/mbc1′ (135) are fundamentally different. These space groups can also be written as a coset decomposition in terms of structurally chiral halving subgroups, which take the form

Each of these halving subgroups hosts a fourfold degenerate “doubled spin-1/2” fermion,^41,42,79 which is phenomenologically distinct from a spin-3/2 fourfold degeneracy (i.e., there are no Chern number 3 bands and no cubic rotational symmetries). Upon the addition of inversion symmetry, these double spin-1/2 degeneracies double again, yielding an eightfold degeneracy, which was aptly named a double-Dirac fermion.⁷⁸ Thus, the tetragonal double Dirac fermions are distinct from Rarita-Schwinger fermions.

We proved in Sec. IV C 1 that every cubic Type II space group with eightfold degeneracies hosted Rarita-Schwinger fermions. The same is true for the Type III magnetic groups. To see this, note that every cubic Type III magnetic space group with an eightfold degeneracy is a subgroup of a Type II space group that also has an eightfold degeneracy. One can thus obtain the k · p Hamiltonian for each Type III group by starting from the RS Hamiltonian in the Type II group and relaxing symmetry constraints. This means that if the RS Hamiltonian is accessible as a point in parameter space for the Type II group, it must also be for the Type III group.

To conclude, we analyze the Type IV magnetic groups that host eightfold degeneracies. We will show below that these do not give RS fermions but are still rather novel. There are three cases given in Table II. Each has a coset decomposition in terms of a Type IV structurally chiral halving subgroup,

In all three cases, the structurally chiral halving subgroup features a fourfold chiral fermion at the R point. Upon the addition of the orientation reversing symmetry (inversion or rotoinversion), the degeneracy doubles, yielding a nonchiral eightfold degenerate fermion. We illustrate this for only the simplest case, $PIm3¯m$⁠, as all three cases are similar. At the R point, the little group of the halving subgroup P_I432 has the following four-dimensional irreducible corepresentation ρ, expressed in the basis introduced in Sec. IV B,

Imposing these symmetry constraints, the fourfold fermion at the R point in P_I432 takes the particularly simple form

which describes a perfectly isotropic doubled spin-1/2 fermion.⁴² Note that the twofold degeneracy at generic k will be lifted by quadratic corrections. Imposing inversion symmetry results in the eight-dimensional corepresentation of the little group in $PIm3¯m$⁠,

Imposing these symmetry constraints on the k · p Hamiltonian H(k), we find

This describes a perfectly isotropic double Dirac fermion, which is fourfold degenerate everywhere at linear order. In particular, this Hamiltonian describes two decoupled doubled spin-1/2 fermions of opposite chirality, with chirality matrix aμ_z − bμ_y. This is distinct from an RS fermion. However, because the doubled spin-1/2 fermions of opposite chirality number are decoupled to linear order, we expect that these double Dirac fermions exhibit an anomalous negative magnetoresistance in analogy to a Dirac semimetal.⁸⁰ 

We now turn to the threefold degenerate fermions listed in Table II. Before describing the magnetic groups, we recall the “spin-1 Weyl” introduced in Ref. 40, described by the local Hamiltonian,

We have omitted an overall scale factor and energy shift that are not determined by symmetry and that do not impact the topological properties. At ϕ = π/2, H_Weyl-1(π/2, k) = k · S, where S_x,y,z are the generators of SO(3) in the spin-1 representation (this explains the nomenclature). H_Weyl-1 has the property that at generic values of ϕ (i.e., ϕ ≠ nπ/3), the bands are only degenerate at k = 0. Thus, we can evaluate the Chern number of each band over a sphere enclosing the degeneracy point; the Chern number is either ±2 or 0 depending on whether the Fermi energy is in the upper/lower bands or the middle band.⁴⁰ 

The spin-1 Hamiltonian in Eq. (57) was derived as the k · p Hamiltonian near the P ≡ (1/4, 1/4, 1/4) point in SG I2₁3 (199). The little group of this point, which we denote $GP199$⁠, is generated by {C_3,111|0} and ${C2z|12012}$⁠, along with the Bravais lattice translations.⁸¹ The spin-1 representation of this group, as given in the BCS,⁷³ is

Now, let us consider the magnetic groups in Table II.

We first consider SGs $P43′32′(212.61)$ and $P41′32′(213.65)$⁠. In these groups, the little group at the threefold degeneracy contains $GP199$ as a halving subgroup,

where $g={C2,110|143434}T$ in $P43′32′$ and $g={C2,110|341414}T$ in $P41′32′$⁠.⁸² These two groups are enantiomorphic partners: they differ only by one generator, which is indicated by the subscript 4₃ or 4₁.

We could derive the representation of G_k from Eq. (58) with g as an additional generator, as we did in Eqs. (14)–(19). However, this work is already done if we consider the time-reversal invariant supergroup $Gk∪TGk$⁠, for which the representations are listed on the BCS.⁷³ The time-reversal invariant supergroup of $P43′32′$ is P4₃321′(212). Its six-dimensional time-reversal invariant little group representation at R, Δ²¹², is given by

This representation is irreducible. However, in the magnetic group $P43′32′$⁠, ${C2,110|143434}$ and $T$ are not separately generators, and only their product (g) is a generator. Since it is possible to simultaneously block diagonalize $Δ212({C3,111|0}),Δ212({C2z|12012})$ and $Δ212(g)=Δ212({C2,110|143434})Δ212(T)$ into 3 × 3 blocks, this representation of $P43′32′$ is reducible. Each 3 × 3 block is itself an irrep of the little cogroup of $P43′32′$⁠, which constitutes a three-dimensional fermion in the magnetic group $P43′32′$⁠. (The two blocks together comprise a sixfold fermion, which is irreducible in the Type II group P4₃321′, as was described in Ref. 40.)

Following the same procedure as in Sec. IV B, the k · p Hamiltonian at $R=(12,12,12)$ is exactly given by the spin-1 Weyl Hamiltonian in Eq. (57); the extra generator $gT$ does not place any further constraint on the Hamiltonian. Thus, in $P43′32′$⁠, the chiral fermion at the R point will exhibit all of the same exotic features—for example, double Fermi arcs and a striking Landau level spectrum—as the threefold fermion in SG I2₁3 that we studied extensively in Ref. 40.

In SG $P41′32′$⁠, the Type II supergroup is P4₁321′(213). The equations are identical to Eq. (60), and hence, $P41′321′$ also hosts a spin-1 Weyl.

We now consider $I41′32′(214.69)$⁠, which is the body-centered generalization of $P43′32′$ and $P41′32′$⁠. It does not have an enantiomorphic partner because it contains both $g={C2,110|341414}T$ and ${C2,110|143434}T$ (here and below, when we write {R|t}, the vector t is written with respect to the conventional cubic lattice vectors, consistent with the notation of the BCS). The little group is the same as in the previous two cases, with the addition of the body-centered translation vectors. Thus, the threefold fermion—now at the P point—is identical to that in I2₁3.

Next, we consider $I4¯′3d′(220.91)$⁠, which has $g={m11¯0|141414}T$⁠.⁸² In this case, g is not in the little group at P; hence, the little group is unitary and thus the low-energy dispersion near P is identical to $GP199$⁠. Although in its parent time-reversal invariant space group, $I4¯3d1′$⁠, line nodes emanate from the threefold degenerate point,⁴⁰ in the magnetic group the bands are nondegenerate away from the threefold degeneracy because the glide symmetry that protected the line nodes (in combination with {C_3,111|0}) is broken down to g.

Finally, we consider the last Type III group with a threefold degeneracy, $Ia3¯d′(230.148)$⁠, which has $g={C2,110|341414}T$⁠.⁸² In this case, g is in the little group at P, and the little group has the same generators as $I41′32′$⁠; consequently, it also displays a threefold degeneracy identical to that in I2₁3.

We now consider the Type IV groups with threefold degeneracies. When considering the Type IV groups, we will not use its Type II supergroup because it does not have the same Brillouin zone, and hence, generically, the little group in the Type II supergroup contains fewer elements. Instead, it turns out to be straightforward to derive the little group representation directly.

We start with SG P_I2₁3(198.11), where Eq. (59) holds with $g={E|121212}T$⁠.⁸² (As is required for a Type IV group, P_I2₁3 contains the product of time reversal and a fractional lattice translation.) We derive the commutation relations