Can spin-1/2 emerge as a property of quasiparticles if original description of the system was without spin? 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.
 A: The answer is Yes. See 
A physical understanding of fractionalization
http://arxiv.org/abs/hep-th/0302201
Quantum order from string-net condensations and origin of light and massless fermions,
    Xiao-Gang Wen;
Spin-1/2 and Fermi statistics from qubits
http://arxiv.org/abs/hep-th/0507118
Quantum ether: photons and electrons from a rotor model,
    Michael Levin, Xiao-Gang Wen;
Spin-1/2 Fermi statistics from rotors
In fact, every lattice QCD or lattice QED is a model where spin-1/2 emerge from something with no spin. But in lattice QCD or lattice QED, Fermi statistics is added by hand.
Not only spin-1/2, almost everything can emerge from interacting qubits.
Wheeler's "it from bit" represents a deep desire to unify matter and information. In fact, it happend before at a small scale. We introduced electric field to informationally (or pictorially) describe Coulomb law. At this stage, the electric field is just information (bit). But later, electric field became real matter with energy and momentum, and even a particle associated with it.
However, in our world, "it" are very complicated. (1) Most "it" are fermions, while "bit" are bosonic. Can fermionic "it" come from bosonic "bit"? (2) Most "it" also carry spin-1/2. Can spin-1/2 arises from "bit"? (3) All "it" interact via a spectial kind of interaction -- gauge interaction. Can "bit" produce gauge interaction? Can "bit" produce waves that satisfy Maxwell equation? Can "bit" produce photon?
More generally, there are Eight wonders in our universe (ie "it" has eight wonders)::


*

*Identical particles.

*Gauge interactions.    

*Fermi statistics 

*spin-1/2 

*Chiral fermions.

*Small mass of fermions. (Much less than Planck mass) 

*Lorentz invariance.

*Gravity.


It turns out that we can only produce the first of eight wonders from bits.
However, if we start with qubits, we can obtain Fermi statistics, spin-1/2, Maxwell equation, Yang-Mills equation, and the corresponding gauge interations. So far, we can  unify seven out of eight wonders (1 -- 7) by qubits, and we are trying to add the gravity (see http://arxiv.org/abs/0907.1203 ).
So It from qubit, not bit.  Bit is too classical to produce Gauge interactions, Fermi statistics, and Chiral fermions. Those phenomena (or properties) come from quantum many-body entanglement, which exists only for qubits.  (See also What is the relationship between string net theory and string / M-theory? )
A: The classic example here is the Thirring model, which describes fermions in 1+1 dimensions. 
While this is a very special two dimensional model (so the conclusions cannot be generalized), Sidney Coleman found that it is equivalent to the Sine-Gordon model, a theory of bosons. The demonstration is rather technical (Coleman essentially proved that the perturbation series for both models are term by term identical) so I won't get into it, but the gist of it is that the Sine Gordon solitons may be identified with the Thirring model fundamental fermions. So the Sine Gordon model is an explicit one dimensional example of what you want: an extended configuration (the soliton) which behaves like a fermion. 
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2088
http://www.sciencedirect.com/science/article/pii/0003491675902122
A: Some of my recent results may be relevant, but they use electromagnetic field as input, and electromagnetic field is associated with spin 1.


*

*Three out of four components of the Dirac spinor function can be algebraically eliminated from the Dirac equation in an arbitrary electromagnetic field. The resulting equation for one complex function (which can be made real by a gauge transform) can be regarded as "spinless", but is generally equivalent to the Dirac equation. This was shown in http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf (JOURNAL OF MATHEMATICAL PHYSICS 52, 082303 (2011)) and, in a much more general and, arguably, attractive form, in http://arxiv.org/abs/1502.02351 .

*After introduction of a complex four-potential of electromagnetic field, which generates the same electromagnetic fields as the initial real four-potential, the spinor
field can be algebraically eliminated from the equations of spinor electrodynamics (the Dirac-Maxwell electrodynamics). The resulting equations for electromagnetic field describe independent evolution of the latter and can be embedded into a quantum field theory. Thus, the modified Maxwell equations describe not only electrodynamic field, but spin 1/2 field as well (http://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-013-2371-4.pdf , Eur. Phys. J. C (2013) 73:2371 ).
