# What is the difference between Bosonic and Fermionic symmetry protected topological phases (SPT)

I am reading the paper Braiding statistics approach to Symmetry Protected Topological Phases'' by Levin and Gu. In this paper two spin models considered describe spin-1/2 particles in (1+2) dimensions, so I would be inclined to believe that these are fermionic particles.

However, the paper refers to the topological phases as bosonic SPTs''. The classification of bosonic and fermionic here is somewhat confusing to me, so question is:

What is the difference between a bosonic SPT and a fermionic SPT?

What I have figured out from reading bits and pieces of other papers is that bosonic SPTs are classified by group cohomology theory, whereas fermionic SPTs are classified by group supercohomology theory.

But why is the spin-1/2 lattice system considered in this paper (by Levin and Gu) a bosonic SPT?

[I should admit that I do not know a lot about this field, and so I will certainly appreciate pointers to background material.]

• In case you're interested in a general awnser about symmetry protection, here is a question I asked a few days ago with a great awnser. – Dimitri May 2 '16 at 12:33

The model of Levin and Gu is built out of products of the spin-1/2 operators $S_i^x$, $S_i^y$ and $S_i^z$ at each site $i$. These operators commute with each other at different sites ($[S_i^\alpha, S_j^\beta] = 0$ for $i \neq j$), which is the reason why we say this model is bosonic, and the SPT phases in this model are bosonic SPT's.
By constrast, a fermionic model would be built out of fermionic operators $a_i$, $a_i^{\dagger}$, which anti-commute at different sites (e.g. $\{ a_i, a_j \} = 0$ for $i \neq j$.) SPT phases in a fermionic model are called fermionic SPT's.
For a more physical viewpoint, note that the Levin and Gu model can be mapped onto a model of hard-core spinless bosonic particles hopping on a lattice, where (for example) spin-up on a given site $i$ corresponds to an unoccupied site, spin-down corresponds to a singly-occupied, and energetic constraints strongly disfavor more than one particle at a time at the same site. Thus, it is essentially a bosonic system.