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.
The SPT order (for both frermionic and bosonic systems) has the following defining properties:
- Distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry.
- However, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation.
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.
- $E_8$ bosonic QH state is not SRE. It is LRE with invTO.
- $\nu=1$ fermionic IQH state is not SRE. It is LRE with invTO.
- $p+ip$ 1D superconducting chain is not SRE. It is LRE with invTO.
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
Yes, this is a known problem in the condensed matter community. There are two different definitions of short range entangled (SRE) states. The key difference is whether the fermion symmetry protected topological (SPT) state belongs to the SRE state. Strictly speaking, the fermion state is not SRE, because fermion statistics is already a phenomenon of long range entanglement, so in this sense, even free fermion state is topologically ordered, and belongs to the long range entangled (LRE) state. Therefore the fermion SPT state should actually be considered as symmetry enriched topological (SET) state. But if we do not insist on this fermion topological order (one may think of quotient out the fermion topological order), we may still ascribe the fermion SPT state to the SRE state, but we should always bare in mind the difference between boson SPT and fermion SPT. In conclusion, in the special sense, symmetric SRE states = boson SPT states, but in the general sense, symmetric SRE states = boson SPT states + fermion SPT states.
The statement that "there cannot be SPT phases with trivial symmetry" is in the special sense, meaning that there is no non-trivial boson SPT state with trivial symmetry. But there can still be non-trivial fermion SPT state without symmetry (because there is indeed this fermion topological order that can not be trivialized), and the 1d Majorana chain is one such example. The fermion SPT state without symmetry is only SRE state in the general sense but not in the special sense, that is why the confusion arises. But there is no physical contradiction here, it is just a matter of different conventions.