Existence of bulk chiral fermions and crystal symmetry

We consider the existence of bulk chiral fermions around points of symmetry in the Brillouin zone of nonmagnetic three-dimensional (3D) crystals with negligible spin-orbit interactions. We use group theory to show that this is possible, but only for a reduced number of space groups and points of symmetry that we tabulate. Moreover, we show that for a handful of space groups the existence of bulk chiral fermions is not only possible but unavoidable, irrespective of the concrete crystal structure. Thus our tables can be used to look for bulk chiral fermions in a specific class of systems, namely, that of nonmagnetic $3\text{D}$ crystals with sufficiently weak spin-orbit coupling. We also discuss the effects of spin-orbit interactions and possible extensions of our approach to Weyl semimetals, crystals with magnetic order, and systems with Dirac points with pseudospin 1 and $3/2$. A simple tight-binding model is used to illustrate some of the issues.

Figure 1

Effects of spin-orbit coupling on the orbital Weyl point of a cubic crystal (top) and graphene (bottom) in arbitrary units.

Figure 2

Bands for the cubic lattice in the NN approximation for $t=-1$. The bcc Brillouin zone with its points and lines of symmetry can be seen in Fig. 3.

Figure 3

Brillouin zone for bcc crystals (Refs. 27 and 28).

Figure 4

Bands for the cubic lattice with a tetragonal distortion ($\epsilon =0.4$) for $t=-1.25$. See the main text for details.