Abstract
We describe the symmetry protected nodal points that can exist in magnetic space groups and show that only three-, six-, and eightfold degeneracies are possible (in addition to the two- and fourfold degeneracies that have already been studied). The three- and sixfold degeneracies are derived from “spin-1” Weyl fermions. The eightfold degeneracies come in different flavors. In particular, we distinguish between eightfold fermions that realize nonchiral “Rarita-Schwinger fermions” and those that can be described as four degenerate Weyl fermions. We list the (magnetic and nonmagnetic) space groups where these exotic fermions can be found. We further show that in several cases, a magnetic translation symmetry pins the Hamiltonian of the multifold fermion to an idealized exactly solvable point that is not achievable in nonmagnetic crystals without fine-tuning. Finally, we present known compounds that may host these fermions and methods for systematically finding more candidate materials.
Highlights
The prediction and observation of Weyl1–8 and Dirac9–13 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 chiral8,22 and gravitational anomalies23,24 and exhibit many novel physical properties, such as gapless Fermi arc surface states,1 extremely large magnetoresistance,25 and giant nonlinear optical response.26,27Topological semimetals 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
In Ref. 40, we completed the catalog of nodal point fermions that exist in systems with time-reversal symmetry and spinorbit coupling by showing that nonsymmorphic symmetries can stabilize three, six, and eightfold degeneracies at the corners of the Brillouin zone and, that chiral fourfold degeneracies are possible
This should not be confused with the usage to refer to symmetry groups with only orientationpreserving 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
Summary
The prediction and observation of Weyl and Dirac fermions catalyzed an intense theoretical and experimental search for topological nodal semimetals in condensed matter systems. These systems provide a solid state realization of chiral and gravitational anomalies and exhibit many novel physical properties, such as gapless Fermi arc surface states, extremely large magnetoresistance, and giant nonlinear optical response.. While “Rarita-Schwinger Weyl” fermions—the chiral halves of RS fermions—have already been predicted to exist in condensed matter systems, 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). 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. In the Type III groups, G′ and G′′ have the same unit cell as G and H, while
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