So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for both the interacting and non-interacting cases.
I understand why we had a $\mathbb{Z}$ classification in the non-interacting case. Basically, it measures the number of dangling majorana modes when one cuts a TR-invariant majorana chain (see section II of the paper). But then, for the interacting case, apparently, in order to define the interacting invariants, and construct the matrix product states, we need to do a Jordan-Wigner transformation to a bosonic spin chain, so that the original hamiltonian is written in terms of the new bosonic variables, $\sigma_j^x$, $\sigma_j^y$,$\sigma_j^z$ (end of section II).
I don't understand why do we need to do this transformation and what good does it do to get the $\mathbb{Z}_8$ classification in the interacting case. I don't see the connection between this transformation and the way he constructed the matrix product states in section III.
Best Answer
There are two ways to understand/derive the $\mathbb Z_8$ different SPTs for fermionic chains with $P$ and $T$ symmetry (or for more general fermionic cases).
1. Map them to bosonic chains and unleash the power of Matrix Product States.
`Matrix Product States' are a very powerful technique for bosonic/spin chains. Basically, as proven by Hastings (2007), any gapped spin chain allows for an MPS representation, which is a certain way of writing the ground state in terms of a tensor network. These tensors have very nice properties. In particular this tensor can be written as a product of tensors, one for every physical site (and for translationally invariant states you have the same tensor on every site). These on-site tensors have three indices: one physical one, and two virtual ones, and the latter two connect the on-site tensor to the tensor of the site to the left and to the tensor on the right. It was then realized by Perez-Garcia, Wolf, Sanz, Verstraete & Cirac (2008) that acting an on-site symmtry (such as spin rotation etc) on the physical index is equivalent to acting a different operator $U$ on the virtual indices. In particular these $U$'s can be proven to form a projective representation of the original symmetry. It was then realized by Fidkowski & Kitaev (2010); Turner, Pollmann & Berg (2010) and Chen, Gu & Wen (2010) (August was a busy month!) that the beauty is that these projective representations then classify all bosonic phases! For example if the on-site symmetry is an spin-$1$ $SO(3)$ symmetry, then the projective representation on the bond is either $SO(3)$ (integer spin) or $SU(2)$ (half-integer spin). The latter case is a non-trivial SPT called the Haldane phase (which was in fact patient zero of the MPS approach!). From this one can actually deduce that if the state has open boundaries, then the edges transform under this projective representation (cf. the spin-$\frac{1}{2}$ edges of the spin-$1$ Haldane phase)
So Matrix Product States allow for a complete and elegant classification of spin chains. This does not directly apply to fermionic systems (there are Grassmannian generalization of MPS, but I don't know whether there are similarly nice results for it). So one approach is to note that any fermionic system with fermionic parity symmetry maps to a local spin chain under Jordan-Wigner, so classifying the fermionic chains then comes down to classifying the spin chains. Conceptually this is note so nice: Jordan-Wigner is a non-local transformation and can change the physics (e.g. as you probably know the single Kitaev chain, which is a symmetry-preserving state, maps to the symmetry broken Ising chain). Nevertheless, Jordan-Wigner preserves the energy spectrum and hence phase transitions, so it is a valid way of in principle seeing how many phases there are (and one has to exercise some care to figure out which of these are symmetry broken and symmetry preserving). This is for example the approach followed by Chen, Gu & Wen in section V.
2. `Fermionic chains matter!'
One can also tackle the fermionic chains on their own right. The gain is conceptual insight (since unlike Jordan-Wigner you respect the physics) and perhaps a faint hope for finding out generalizations to higher dimensions (?), and the loss is the undeniable power of MPS. In particular Matrix Product States can be efficiently obtained using numerical methods (like DMRG), which means that if one puts in the Hamiltonian one can easily calculate what kind of symmetry broken state or SPT the spin chain is. For fermionic chains --if one does not opt for option (1)-- one has to use more basic methods. However, I personally quite like this, as it shows that MPS is not the conceptual explanation for SPTs, but rather a (beautiful and very useful) tool for computing what SPT one is in.
And even then there are two ways of doing it in the fermionic setting: by focusing on entanglement properties, or on edge properties. But in fact, these are usually equivalent, both mathematically and intuitively, so it is usually a matter of taste and convenience. The above cited work by Turner, Pollmann and Berg works out the $\mathbb Z_8$ classification in the fermionic setting in terms of the entanglement language. As far as I can see, Fidkowski & Kitaev do a bit of a hybrid, sometimes discussing it in terms of the corresponding spin language, and sometimes sticking to the purely fermionic case. As I was recently figuring this stuff out as well, I have written some notes for myself on how to understand and calculate these different SPTs by focusing on the edge state behaviour (both in the fermionic and bosonic case) without using MPS. Of course everything is there in the articles above, but if it can help to read a small review on the level of a PhD student, I would gladly share! So let me know if there is any interest (to see an example of the idea, you can see my answer here).
EDIT: in a recent paper, I attempt to give an accessible review of the classification without appealing to MPS, in line with I described above. (In section I, I give a bit of an overview, mostly relying on examples to get the message across. In the Appendix I then go through things more systematically.)