How does the spin-statistics theorem apply in condensed matter?

The spin-statistics theorem is an important result in quantum field theory. It shows that particles with integer spin must be bosons and particles with half-integer spin must be fermions.

Confusion then arises in the quantum many-body theory of condensed matter because there are theories [and materials!] which involve quasi-particles which appear to violate this theorem. Here are some examples:

• Spinless fermions. These arise in one-dimensional models. For example, the transverse field Ising model.
• Schwinger bosons which are spinors (bosonic spinons). These arise in Sp(N) representations of frustrated quantum antiferromagnets. They were introduced by Read and Sachdev.
• Anyons. In two dimensions one can have quasi-particles which obey neither bose nor Fermi statistics.

How is this possible? Like a lot of inconsistencies, the answer is to look at the underlying assumptions required to prove the theorem. These include assuming:

1. The theory has a Lorentz invariant Lagrangian.
2. The vacuum is Lorentz invariant.
3. The particle is a localized excitation. Microscopically, it is not attached to a string or domain wall.
4. The particle is propagating, meaning that it has a finite, not infinite, mass.
5. The particle is a real excitation, meaning that states containing this particle have a positive definite norm.

I think three dimensions [and a non-interacting, i.e. quadratic Hamiltonian] may be other assumptions.

In condensed matter, one or more of the above assumptions may not hold. For example, 

• inclusion of a discrete lattice breaks Galilean invariance
• spontaneous symmetry breaking  
• topological order can lead to non-local excitations
• in one dimension spinless fermions may be non-local [e.g. associated with a domain wall]

Thanks to Ben Powell for reminding me of this question.