When studying Symmetry Protected Topological phases, one needs to define what a short range entangled (SRE) states means. But there appears to be different definitions that are not equivalent to each other. In http://arxiv.org/abs/1106.4772, Xiao-Gang Wen defined SRE states to be a state that can be transformed into the unentangled state (direct-product state) through a local unitary evolution. This implies in particular, that there cannot be SPT phases with trivial symmetry, because states with trivial symmetry can always be unitarily evolved to a product state. This is apparently contradicted to Kitaev's notation of SRE. In http://arxiv.org/abs/1008.4138, Kitaev said that there can be non-trivial SPT phases for a Majorana chain with trivial symmetry in 1+1d characterized by dangling Majorana modes at the two ends. My question is, what is Kitaev's definition of SRE (I cannot find a reference where Kitaev explicitly defined this), and how is it differed from Wen's definition. Apparently, If a state is SRE in Wen's definition, then it is SRE in Kitaev's definition.
[Physics] Definition of short range entanglement
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Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?"
(1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-reversal and U(1) symmetries by definition). Topological superconductor are not p+ip superconductors in 2+1D. But it can be p-wave superconductors in 1+1D.
(2) 1+1D topological superconductor is a SET state with a Majorana-zero-mode at the chain end. But time reversal symmetry is not important. Even if we break the time reversal symmetry, the Majorana-zero-mode still appear at chain end. In higher dimensions, topological superconductors have no topological order. So they cannot be SET states.
(3) In higher dimensions, topological superconductors are SPT states.
The terminology is very confusing in literature:
(1) Topological insulator has trivial topological order, while topological superconductors have topological order in 1+1D and no topological order in higher dimensions.
(2) 3+1D s-wave superconductors (or text-book s-wave superconductors which do not have dynamical U(1) gauge field) have no topological order, while 3+1D real-life s-wave superconductors with dynamical U(1) gauge field have a Z2 topological order. So 3+1D real-life topological superconductors (with dynamical U(1) gauge field and time reversal symmetry) are SET states.
(3) p+ip BCS superconductor in 2+1D (without dynamical U(1) gauge field) has a non-trivial topological order (ie LRE) as defined by local unitary (LU) transformations. Even nu=1 IQH state has a non-trivial topological order (LRE) as defined by LU transformations. Majorana chain is also LRE (ie topologically ordered). Kitaev does not use LU transformation to define LRE, which leads to different definition of LRE.
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.)
Best Answer
A quote from http://en.wikipedia.org/wiki/Symmetry_protected_topological_order :
The SPT order (for both frermionic and bosonic systems) has the following defining properties:
The above also defines short-range entanglement (SRE): A SRE state is a gapped state that can be smoothly deformed into the trivial product state without a phase transition (all the symmetries are allowed to break during the deformation).
This is the original definition given in arXiv:1004.3835 Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen Phys. Rev. B 82, 155138 (2010)
SRE states are trivial. They all have the property of unique ground state on any closed space manifold. So SPT order is actually symmetry protected trivial order (ie symmetry protected SRE order), instead of symmetry protected topological order. (In fact, I agreed to use the name SPT in arXiv:0903.1069 because SPT can stand for both symmetry protected trivial and symmetry protected topological).
Kitaev later gave another definition in a talk Toward Topological Classification of Phases with Short-range Entanglement, 2011. http://online.kitp.ucsb.edu/online/topomat11/kitaev/ : A SRE state is a gapped state with unique ground state on any closed space manifold. We do not call such a state SRE state, but call it invertible topologically ordered (invTO) state. (see
arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions Liang Kong, Xiao-Gang Wen,
arXiv:1406.7278 Short-range entanglement and invertible field theories Daniel S. Freed ).
Kitaev also call his version of SRE as locally definable which may be a better name.
Some examples:
Those states are non-trivial even without symmetry protection. So they are topologically ordered and long-range entangled (LRE). The above states are SRE in Kitaev's sense, but LRE in our sense. Many more examples can be found in arXiv:1406.7329 Fermionic Symmetry Protected Topological Phases and Cobordisms Anton Kapustin, Ryan Thorngren, Alex Turzillo, Zitao Wang