I am currently reading a review "Area law for the entanglement entropy" by Eisert, Cramer and Plenio (2010). From what I understand:
- In one dimension, for local gapped models, we have an area law for entanglement entropy.
- In one dimension, some models with long range interactions + all critical systems obey a log(N) law, where N is the size of the subregion of the 1D space, the pre-factor depends for CFT models on the central charge.
- For higher dimensions, local gapped (equivalently quasi-free) models obey an area law.
- For higher dimensions, critical bosonic models obey an area law, fermionic ones obey a divergent log law with pre-factor dependent on the topology of the Fermi surface (which is also probably related to the central charge of the CFT, but no exact results are known).
However I am a bit confused about the treatement of the Klein-Gordon field.
The massive Klein-Gordon field is a local gapped model, hence not critical. Papers by Bombelli and Srednicki show that entanglement entropy obeys an area law for dimension greater than 2, in agreement with point 3. However in the review they seem to say that this is a critical system. Thanks to point 4 there is still no contradiction with the area law result.
However, this becomes problematic for the 1D case: a massive or massless KG system obeys a divergent log law, as shown here: https://arxiv.org/abs/hep-th/9401072 and in the review. To me this is only true if the field is massless. A massive KG field is a local gapped system, so not critical and hence it should follow an area law according to point 1, not a divergent log law.
So it seems to me that the problem is that these papers consider the KG field (massive or massless) as critical, whereas to me only a massless KG field is critical.
What did I misunderstand?