I have a question about entanglement in condensed matter physics. It seems that topological order origins from long range entanglement, but what is long range entanglement? It is the same as long range correlation? I am interested in this issue and I am happy to have any discussion.
Long range entanglements are defined through local unitary transformations which is discussed in arXiv:1004.3835 Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order by Chen, Gu, and Wen.
Basically, long range entangled states are states which are very different from product states and cannot be changed to product states through local unitary transformations.
I think I can give a more technical and detailed explanation of "long range entanglement". I felt it puzzling some time before and still puzzle me for some situations.
For generic topological states, the entanglement of ground states scales as $S=\alpha L-ln(D)$ ($D$ is the total quantum dimension of the system. For topologically nontrivial system, $D>1$) for the reduced density matrix $\rho_A$ whose perimeter is $L$. Good references of this point should be Preskill & Kitaev's paper and Levin & Wen's paper. If we start and keep on coarse-graining the wave function, i.e., block several lattice sites into one, we still end up with an entangled state. The reason is the following: Since we need $S>0$ all the time during our coarse-graining, $\alpha$ cannot be zero. Therefore, our state is still entangled no matter how you do the coarse-graining. The "long range" is in the sense of coarse-graining. For a topologically trivial state, $S=\alpha L$, the above reasoning does not apply here. So we will "probably" end up with a product state with $\alpha=0$ after coarse-graining.
Hope it helps!!