Graphene
Graphene is a material that has been of increasing interest for its applicability to technology. It consists of a single sheet of carbon atoms in a regular hexagonal lattice, with an electron free to move from atom to atom, at vertices of the lattice.
It is relevant for the current topic because its energy levels obey the Dirac equation for a massless particle. This is particularly surprising because the energies involved are not relativistic, which is the realm in which the Dirac equation applies on a fundamental level. Instead it appears that the massless 2-dimensional Dirac equation emerges as a property of motion on the hexagonal lattice.
Paper on spin and graphene includes this quote in the Abstract:
Furthermore, it suggests that the half-integer spin of the quarks and leptons could derive from hidden substructure, not of the particles themselves, but rather of the space in which these particles live. In other words, the existence of spin might be interpreted as evidence that space consists of discrete points arranged in a non-cubic lattice.
This review paper looks very good and I am reading it now