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.]

  • $\begingroup$ 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. $\endgroup$
    – Dimitri
    Commented May 2, 2016 at 12:33

1 Answer 1


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.

On the other hand, as you alluded to in your question, it is also true that the Levin and Gu model can be realized in a system of spin-1/2 fermions at half-filling, where there is a large energy penalty for more than one fermion on a given site so that the fermions can't move ("Mott insulator") and the low-energy physics is just the interactions between the spins. This is an example of a more general principle: Bosonic systems are a subset of fermionic systems. This is because we can always add interaction terms such that at low energies the fermions pair into bosons, and then work with the bosonic variables.

Thus, we see that bosonic SPT's are essentially a subset of fermionic SPT's*. For example, consider the classification of fermion topological phases in (1+1)-D with time-reversal symmetry, http://arxiv.org/abs/1008.4138. There are a total of 8 phases (including the trivial one), which we can label as {0,1,2,3,4,5,6,7}. Of these, 0 and 4 are essentially bosonic in character (because 0 is the trivial phase in any case, and 4 is characterized by the presence of a Kramers doublet on the boundary, which can equally well occur in a bosonic system.) The rest are truly fermionic phases which have no analog in bosonic systems. (For example, 1,3,5,7 have a Majorana fermion zero mode on the boundary, and 2 and 6 have an action of time-reversal which changes the fermion parity on the boundary.)

*There is one caveat to the statement that bosonic SPT's are a subset of fermionic SPT's. The reason is that a system which is in a non-trivial bosonic SPT phase (cannot be continuously connected to the trivial phase through a continuous path of bosonic Hamiltonians without crossing a phase transition) can become trivial as a fermionic SPT if there exists a continuous path of fermionic Hamiltonians (which, recall, are more general than bosonic Hamiltonians) connecting it to the trivial phase. Some examples of this phenomenon can be found in: http://arxiv.org/abs/1205.3156


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