Sixfold Fermions: Symmetry, Topology & Materials

• Sixfold fermions are low-energy quasiparticles arising from sixfold-degenerate Bloch-band crossings enforced by crystal symmetry instead of Lorentz invariance.
• They manifest in distinct symmetry settings—either as chiral double spin-1 nodes with nonzero Chern numbers or as nonchiral triple Dirac nodes with quadratic dispersion.
• Their realization in materials like AlPt, PtBi₂, and C12A7:4e− underpins novel transport phenomena, magnetic-field responses, and catalytic activity via topological Fermi arcs.

Searching arXiv for recent and foundational papers on sixfold fermions to ground the article in published work. Sixfold fermions are low-energy quasiparticles realized at sixfold-degenerate Bloch-band crossings whose existence is fixed by crystal symmetry rather than by Lorentz invariance. In the current literature, they occur in several distinct symmetry settings: as chiral sixfold nodes in chiral space-group-198 materials, commonly interpretable as time-reversal-doubled spin-1 fermions with large Chern number; as nonchiral “triple Dirac” nodes in centrosymmetric pyrite systems, where a three-dimensional little-group irrep is doubled by inversion and time-reversal symmetry; and as multifold nodes generated by interstitial-electron bands in electrides such as C12A7:4e− (Schröter et al., 2018, Barman et al., 2020, Shen et al., 2023, Thirupathaiah et al., 2020, Yáng et al., 2020, Meng et al., 2021).

1. Conceptual classification

Sixfold fermions belong to the broader class of multifold fermions, namely crystalline quasiparticles associated with band crossings of degeneracy higher than the twofold Weyl and fourfold Dirac cases. The central organizing principle is the little group at a high-symmetry momentum: if the little group admits a three-dimensional irrep, then spinful time-reversal-invariant settings can promote that crossing to a sixfold degeneracy by symmetry-enforced doubling. This is why sixfold fermions are regularly described either as “double spin-1” excitations in chiral crystals or as “triple Dirac points” in centrosymmetric crystals (Barman et al., 2020, Thirupathaiah et al., 2020, Bavaro et al., 15 Jul 2025).

The literature distinguishes at least three non-equivalent realizations. In chiral, non-centrosymmetric crystals such as AlPt and the SrGePt family, the sixfold node at $R$ is tied to spin-orbit coupling and time-reversal doubling of a spin-1 object, and it can carry a nonzero net chiral charge of magnitude $4$ (Schröter et al., 2018, Shen et al., 2023). In centrosymmetric Pa-3 systems such as PtBi$_2$ and PdSb$_2$, the sixfold node is nonchiral because inversion and time-reversal enforce vanishing Berry curvature or vanishing net Chern number at zero field (Thirupathaiah et al., 2020, Yáng et al., 2020). In C12A7:4e−, the sixfold point is again nontrivial but its microscopic origin is unusual: it derives from floating interstitial-electron bands rather than conventional atom-centered orbitals (Meng et al., 2021).

A recurring misconception is that sixfold fermions form a single universal class with a single dispersion law and a single surface-state phenomenology. The material record does not support that simplification. Some sixfold nodes are linear and chiral, some are quadratic and nonchiral, and some generate exceptionally long Fermi arcs while others do not produce isolated arc phenomenology at all (Schröter et al., 2018, Thirupathaiah et al., 2020, Yáng et al., 2020).

2. Symmetry origin and representation theory

In chiral space group 198, the sixfold node occurs at the Brillouin-zone corner $R$ in the spinful case. The protecting algebra is built from non-symmorphic twofold screws and a threefold rotation about $(111)$, together with time-reversal symmetry. At $R$, the relevant spinful relations differ qualitatively from the spinless case: the screws commute and square to $+1$, while the threefold operation generates a three-dimensional irrep; time reversal then produces three additional orthogonal partners, yielding a symmetry-enforced sixfold degeneracy (Barman et al., 2020). AlPt realizes this structure in the chiral cubic space group $P2_13$ (No. 198), where spin-orbit coupling and time-reversal symmetry turn the spin-1 crossing at $R$ into a sixfold node (Schröter et al., 2018).

The SrGePt family provides a closely related but not identical realization in space group $4$0 (No. 198). There, the little group at $4$1 contains a threefold rotation along $4$2, twofold screw symmetries $4$3 and $4$4, and time-reversal symmetry. The sixfold crossing is explicitly described not as a single six-dimensional irrep, but as the time-reversal doubling of a triply degenerate spin-1 fermion, formed from two copies of the $4$5 irrep at the time-reversal-invariant momentum $4$6 (Shen et al., 2023).

In centrosymmetric pyrite systems with space group Pa-3 (No. 205), the symmetry mechanism is different. In cubic PtBi$4$7, the little group at the relevant $4$8 point contains a threefold screw $4$9 along $_2$0, a twofold screw $_2$1 along a principal axis, inversion $_2$2, and time-reversal symmetry $_2$3. The orbital sector realizes a three-dimensional irrep, and spin together with $_2$4 and $_2$5 produces the sixfold node (Thirupathaiah et al., 2020). In $_2$6-PtBi$_2$7, the same space group and the same $_2$8 corner yield a three-dimensional double-little-group irrep which is doubled by inversion and time-reversal symmetry into a sixfold-degenerate touching (Bavaro et al., 15 Jul 2025). PdSb$_2$9, also Pa-3, is described analogously: the nonsymmorphic little group at $_2$0 supports a three-dimensional irrep which becomes sixfold in the spinful, inversion- and time-reversal-symmetric setting (Yáng et al., 2020, Chapai et al., 2019).

C12A7:4e− introduces a separate symmetry setting. The crystal belongs to the cubic space group $_2$1-43d (No. 220), and the sixfold-degenerate point appears at the bulk $_2$2 point. Its protection is tied to non-symmorphic symmetries in the $_2$3-point little group, including $_2$4, additional screw or glide elements, and a threefold rotation $_2$5. Along $_2$6–$_2$7, the sixfold point splits into three doubly degenerate nodal lines protected by the glide mirror $_2$8 and time-reversal symmetry (Meng et al., 2021).

3. Effective Hamiltonians, dispersion, and topology

The simplest analytical picture of a chiral sixfold fermion is the time-reversal-doubled spin-1 model. In space-group-198 systems, the underlying threefold node can be written as

$_2$9

with $R$0 the $R$1 spin-1 generators. The sixfold node is then represented as

$R$2

or, in AlPt, equivalently as $R$3. This yields three Kramers-degenerate branches corresponding to $R$4, with Chern numbers $R$5 for the doubly degenerate bands in the ideal double-spin-1 limit (Schröter et al., 2018, Barman et al., 2020). In SrGePt, the corresponding $R$6 $R$7 theory is linear in $R$8, anisotropic, and symmetry-constrained by $R$9, $(111)$0, $(111)$1, and $(111)$2; the resulting sixfold node carries $(111)$3 (Shen et al., 2023).

The topological characterization is conventionally expressed through the Berry curvature and the Chern number

$(111)$4

evaluated on a small closed surface enclosing the node. In chiral sixfold systems such as AlPt, CoGe, and SrGePt, this integral yields a nonzero charge of magnitude $(111)$5 (Schröter et al., 2018, Barman et al., 2020, Shen et al., 2023). In CoGe, Wilson-loop calculations on maximally localized Wannier functions and Berry-curvature maps show that the sixfold node at $(111)$6 carries the opposite chiral charge to the spin-3/2 node at $(111)$7, as required by Nielsen–Ninomiya cancellation (Barman et al., 2020).

Centrosymmetric sixfold fermions follow a different low-energy structure. In cubic PtBi$(111)$8, the minimal $(111)$9 Hamiltonian $R$0 is quadratic in momentum about $R$1, with explicit symmetry-constrained orbital blocks and Kramers doubling. Because both inversion and time reversal are present, the Berry curvature vanishes at zero field and the node carries no net monopole charge. A Zeeman field along $R$2 breaks time reversal and splits the sixfold crossing into twenty type-II Weyl cones, eight on the $R$3 axis and twelve on the principal axes, with chiralities determined by Berry-flux integration on small spheres (Thirupathaiah et al., 2020). PdSb$R$4 is described by the same qualitative rule: inversion and time-reversal symmetry forbid linear terms at leading order, so the sixfold node at $R$5 is quadratic and nonchiral (Yáng et al., 2020, Chapai et al., 2019).

C12A7:4e− again constitutes a distinct case. Its sixfold $R$6-point Hamiltonian is linear in $R$7 and block-structured in $R$8 submatrices. In the regime $R$9 greater than the inter-block couplings, the node is interpretable as a composition of two spin-1 fermions with vanishing total monopole charge. The paper therefore assigns nontrivial Berry curvature to individual bands but a zero net topological charge to the sixfold node as a whole (Meng et al., 2021). This shows that “double spin-1” is not synonymous with nonzero node charge; the symmetry-allowed inter-block structure matters.

4. Materials platforms and electronic structure

The known materials record already spans chiral metals, pyrite semimetals, and an electride catalyst.

System	Space group and node	Sixfold characteristics
AlPt	$+1$0 (No. 198), $+1$1	Chiral double spin-1, $+1$2
CoGe / BiSbPt / KMgBO$+1$3	$+1$4 (No. 198), $+1$5	Sixfold chiral node with giant arcs
SrGePt family	$+1$6 (No. 198), $+1$7	SOC-enabled sixfold, $+1$8
PtBi$+1$9 / $P2_13$0-PtBi$P2_13$1	Pa-3 (No. 205), $P2_13$2	Nonchiral triple-Dirac sixfold near $P2_13$3
PdSb$P2_13$4	Pa-3 (No. 205), $P2_13$5	Quadratic nonchiral sixfold
C12A7:4e−	$P2_13$6-43d (No. 220), $P2_13$7	Interstitial-electron sixfold near $P2_13$8

Energy alignment relative to the Fermi level varies substantially across platforms. In AlPt, the sixfold node at $P2_13$9 is observed at binding energy $R$0 eV below $R$1, while the fourfold node at $R$2 lies near $R$3 (Schröter et al., 2018). In SrGePt, the sixfold node sits above $R$4, with $R$5 eV in SrGePt and family values ranging from $R$6 eV in BaSiPt to $R$7 eV in SrSiPd (Shen et al., 2023). In cubic PtBi$R$8, density functional theory places the node approximately $R$9 meV above $4$00, whereas ARPES locates it $4$01–$4$02 meV below $4$03 (Thirupathaiah et al., 2020). In $4$04-PtBi$4$05, the combined quantum-oscillation and DFT analysis favors a node approximately $4$06–$4$07 meV below $4$08 (Bavaro et al., 15 Jul 2025). In PdSb$4$09, the sixfold crossing sits $4$10–$4$11 eV below $4$12 (Yáng et al., 2020). In C12A7:4e−, the sixfold point at $4$13 and the accompanying fourfold point at $4$14 both lie very close to $4$15 (Meng et al., 2021).

The orbital content is equally diverse. In $4$16-PtBi$4$17, the low-energy bands, including those that form the sixfold node, derive from hybridized Pt-$4$18 and Bi-$4$19 states (Bavaro et al., 15 Jul 2025). In PdSb$4$20, the density of states near $4$21 is globally dominated by Sb contributions, but the two lightest bands at $4$22 that host the small extremal pocket probed by de Haas–van Alphen oscillations have stronger Pd character (Chapai et al., 2019). C12A7:4e− is qualitatively different: its excess electrons reside predominantly at the interstitial $4$23 Wyckoff site, the corresponding floating bands are classified as the elementary band representation $4$24, and the sixfold and fourfold nodes inherit that interstitial-electron origin (Meng et al., 2021).

5. Surface states and spectroscopic manifestations

In chiral sixfold systems, the most visible signature is often the presence of long topological Fermi arcs. AlPt provides the canonical example. On the (001) surface, vacuum-ultraviolet ARPES and slab calculations reveal long, spin-split, S-shaped arcs that span the full diagonal of the surface Brillouin zone, connecting $4$25 and $4$26. Their dispersion chirality determines the sign configuration of the bulk node charges and therefore the structural handedness of the crystal (Schröter et al., 2018).

Surface calculations in other space-group-198 compounds reinforce the same pattern while emphasizing material dependence. In CoGe and BiSbPt, (001)-surface spectra display giant arcs or multiple pairs of surface states that connect projected multifold nodes; in KMgBO$4$27, the arcs are especially clean because trivial bulk Fermi pockets are nearly absent (Barman et al., 2020). By contrast, in SrGePt the (010)-surface calculation shows that the sixfold-$4$28-derived features at $4$29 are not cleanly resolved because they overlap strongly with trivial states, whereas the $4$30-derived Rarita–Schwinger–Weyl arcs remain conspicuous (Shen et al., 2023). The existence of a large node charge therefore does not guarantee a spectroscopically isolated sixfold-derived arc on every termination.

C12A7:4e− provides an electride-specific variant of the arc phenomenology. The (001) surface spectral function contains two families of topological Fermi arcs: dumbbell-like arcs associated with the sixfold point projected from $4$31, and petal-like arcs associated with the fourfold point projected from $4$32. Both are exceptionally long, with the $4$33-derived arcs nearly traversing the entire surface Brillouin zone (Meng et al., 2021).

Centrosymmetric Pa-3 systems require a different reading of ARPES data because their zero-field sixfold nodes are nonchiral. In cubic PtBi$4$34, ARPES maps the band structure in good agreement with DFT, locating the sixfold crossing near $4$35, but the experiment also shows that low $4$36 resolution smears the point-like crossing; the observed surface-related “brackets” and “petals” in zero field are not Weyl Fermi arcs (Thirupathaiah et al., 2020). In PdSb$4$37, ultrahigh-resolution ARPES directly resolves three doubly degenerate branches meeting at a single point at $4$38, and the inner branch is accurately fit by a pure quadratic dispersion, establishing the symmetry-forbidden character of linear terms near the node (Yáng et al., 2020).

6. Field response, transport, and functional consequences

Once the sixfold node lies close to the Fermi level, it can control bulk response functions. In cubic PtBi$4$39, the most striking prediction is magnetic-field-induced fragmentation: a Zeeman field along $4$40 transforms the zero-field sixfold fermion into twenty type-II Weyl cones, with eight nodes on the $4$41 axis and twelve on the principal axes. The same work identifies this system as a platform for transport beyond the Dirac and Weyl paradigms, including unusual quantum oscillations, nonlinear Hall and Nernst responses, and field-induced surface Fermi arcs that remain to be mapped experimentally (Thirupathaiah et al., 2020).

Bulk quantum oscillations have now provided direct evidence for sixfold-derived Fermi-surface pockets in $4$42-PtBi$4$43. For $4$44 at $4$45 K, the observed low frequencies are $4$46, $4$47, $4$48, $4$49, and $4$50 T, and they are assigned to three tiny electron pockets centered on the sixfold point at $4$51. The corresponding effective cyclotron masses are $4$52, $4$53, $4$54, $4$55, and $4$56 $4$57. Dingle temperatures of $4$58 K and $4$59 K for representative orbits yield quantum mobilities of approximately $4$60 and $4$61 cm$4$62V$4$63s$4$64, firmly tying ultralight carriers to the sixfold node lying about $4$65 meV below $4$66 (Bavaro et al., 15 Jul 2025).

PdSb$4$67 shows a different response profile. At ambient pressure it is a diamagnetic Fermi-liquid metal with transverse magnetoresistance reaching approximately $4$68 at $4$69 K and $4$70 T, strict Kohler scaling, a hole carrier density of approximately $4$71 cm$4$72, and Hall mobility of approximately $4$73 cm$4$74V$4$75s$4$76 at $4$77 K. For $4$78, de Haas–van Alphen oscillations yield a single frequency $4$79 T, an effective mass $4$80, and a Berry phase consistent with $4$81. Under quasi-hydrostatic pressure, the light $4$82-point bands are pushed above $4$83, and superconductivity emerges above approximately $4$84 GPa, with a dome-shaped $4$85 reaching $4$86 K near $4$87 GPa and $4$88 T at $4$89 GPa (Chapai et al., 2019).

In the chiral SrGePt family, the sixfold node contributes to a large intrinsic spin Hall conductivity through pronounced spin Berry curvature near degenerate nodes and through SOC-induced band splitting. For SrGePt at $4$90, the nonzero tensor components are $4$91 and $4$92 in units of $4$93. Shifting the chemical potential to the sixfold-node energy $4$94 eV changes these values to $4$95 and $4$96, illustrating that the response is energy selective and sensitive to the multifold spectrum near $4$97 (Shen et al., 2023).

C12A7:4e− extends the consequences of sixfold physics into catalysis. On the (001) surface, the hydrogen-adsorption free energy is $4$98 eV, better than NbP $4$99 eV), TaAs $_2$00 eV), and NbAs $_2$01 eV), while adsorption inside the bulk cages is much less favorable with $_2$02 eV. Artificial hole doping, hydrostatic strain, and comparison to Li$_2$03Al$_2$04Si$_2$05:5e− show that moving the sixfold- and fourfold-derived Fermi arcs away from $_2$06 increases $_2$07, reaching $_2$08 eV in the control electride where the multifold nodes lie at least $_2$09 eV above $_2$10 (Meng et al., 2021). This establishes, within that material class, that it is the energetic alignment of near-$_2$11 arc states rather than the mere presence of excess electrons that governs the functional response.

Across these platforms, sixfold fermions are therefore not a single phenomenological entity but a symmetry family whose physical consequences depend on chirality, inversion, time-reversal structure, and energetic placement relative to $_2$12. Chiral sixfolds can generate $_2$13 charges and giant arcs; centrosymmetric sixfolds provide nonchiral triple-Dirac physics and field-induced Weyl fragmentation; electride sixfolds show that even zero-net-charge multifold nodes can dominate surface chemistry when their arc states are brought to the Fermi level (Schröter et al., 2018, Thirupathaiah et al., 2020, Meng et al., 2021).