When we consider a band structure of some crystal, we can get a model of particle-antiparticle system like electrons and holes. In graphene, for instance, we even get a model of massless Dirac fermions. But as I understand, the spin in graphene description by Dirac equation appears there from the beginning — from real electrons — and remains in the effective Hamiltonian.
Can we start from spinless real (possibly interacting) particles (i.e. not quasiparticles) and in some way still get an effective Hamiltonian, which would describe quasiparticles with spin $\frac12$?
Even better, can there be such a periodic potential that some pair of bands or even one band had an effective Hamiltonian, which would look like a Hamiltonian of a spinful particle?
In other words, can we model spinful particles without putting the spin itself originally into the description by hand?
I'm looking for some more or less common description like e.g. Schrödinger's equation, where we would start with spinless description and obtain a spinful effective theory to describe some quasiparticles. A spinful theory like e.g. Dirac's equation, from which spin was eliminated just to be reintroduced back, is not considered a good suggestion.