# Definition of short range entanglement

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:

1. Distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry.
2. 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.

Some examples:

1. $E_8$ bosonic QH state is not SRE. It is LRE with invTO.
2. $\nu=1$ fermionic IQH state is not SRE. It is LRE with invTO.
3. $p+ip$ 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

• As Zitao Wang pointed out that "Kitaev said that there can be non-trivial SPT phases for a Majorana chain with trivial symmetry", which is equivalent to say "there can be non-trivial symmetry protect topological phases with no symmetry", if one uses Kiteav's definition of SRE and if one defines SPT states as SRE states. That sounds strange: where is the symmetry protection without symmetry. So if one defines SPT states as SRE states, then it is better to use our definition of SRE. – Xiao-Gang Wen Sep 19 '14 at 2:08
• It seems that one can also define a Kitaev SRE state to be a state with zero topological entanglement entropy(TEE). The topological entanglement entropy $\gamma$ is defined for d=2+1 in arxiv.org/abs/hep-th/0510092. Vanishing $\gamma$ implies that the ground state is non-degenerate, for a gapped system, this implies that the effective theory is invertible, which agrees with your description of Kitaev SRE. But TEE is only defined for d=2+1, and I cannot find a general definition in other dimensions, so I'm not sure whether the above definition (in terms of TEE) holds in other dimensions. – Zitao Wang Sep 19 '14 at 2:51
• In the field theory language, group cohomology method captures pure gauge theories, however, SRE phases may contain couplings to gravity. Besides, in dims $4k-1$, the effective action can also have the so-called gravitational CS terms that depends explicitly on the metric (the whole eff. action remains topological however), which describes thermal hall conductance. Both of them are included as SRE phases in the Freed paper you cited above. Is it automatic in your definition that SRE implies vanishing thermal hall conductance? Otherwise it should appear in the group cohomology. – Zitao Wang Sep 19 '14 at 3:56
• Yes. My version of SRE implies vanishing thermal hall conductance in 2+1D. – Xiao-Gang Wen Sep 19 '14 at 4:06
• Yes. it can be proved in 2+1d using the arguments in physics.stackexchange.com/questions/135673/… (the thermal hall conductance is quantized, hence must be constant when the Hamiltonian is continuously connected to the Hamiltonian of the trivial state, i.e., equals to 0). However, this seems to be a nontrivial statements in other dimensions. I'll see if I can prove the claim in general. – Zitao Wang Sep 19 '14 at 4:23

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.