# Entanglement entopy and area law

Hi all!

I am currently reading a review "Area law for the entanglement entropy" by Eisert, Cramer and Plenio (2010). From what I understand:

1. In one dimension, for local gapped models, we have an area law for entanglement entropy.
2. 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.
3. For higher dimensions, local gapped (equivalently quasi-free) models obey an area law.
4. 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: http://arxiv.org/pdf/hep-th/9401072.pdf 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?

## Answers and Replies

atyy
Science Advisor
Does the entanglement of a massive Klein-Gordon field really diverge logarithmically?

I think Cardy and Calabrese give an area law for a 1+1 dimensional massive free field in Eq 68 of http://arxiv.org/abs/hep-th/0405152 .

Hmm, it seems you are right. However:

1. In http://arxiv.org/pdf/0808.3773v4.pdf I am referring to equation 10, where the logarithmic negativity diverges with the system size, independently of the field being massive or not. True it is only an upper bound on entropy, but if it diverges it means entanglement is infinite so I guess entropy should also be infinite. This is confirmed by equation 28 in http://arxiv.org/pdf/hep-th/9401072.pdf. The confusion about what is considered as a critical system stems from the remark in the review just above equation 9 (p.6): "we consider an important model for which the energy gap vanishes" (reference to the KG field), whereas this gap doesn't vanish (a-2|b|=dE²=m²)!

2. p .15 they claim that the area law for the KG fields in d>1 shows that the relation between criticality and entanglement divergence is not valid anymore for bosonic fields. So they consider the KG field as critical independently of it being massive or not.

All this adds to my confusion, unless I am missing some (different enough) hypothesis made in each paper.

EDIT: The claim that equation 28 confirms that entropy should diverge is false, since I didn't notice they took the massless limit just above at equation 24. However, I think that my point about entropy being infinite if log negativity is infinite should still be valid. So do the polemic points in the review p.6 and p.15.

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atyy
Science Advisor
Could it be that the logarithmic negativity doesn't indicate the difference in entanglement entropy between critical and non-critical cases? In http://arxiv.org/abs/0811.1288 which has some of the same authors as the review, they say in the second paragraph on p2 "As d → 0 the entanglement in both critical and noncritical fields exhibits a similar behaviour since in the noncritical regime (finite l), ELN diverges as a power law as well." What is confusing is that in the review they seem to say that the massive KG field as critical - but maybe that was an oversight due to the logarithmic negativity being the same for critical and non-critical cases when d → 0? In this paper they do say that the critical case is m → 0 (top right of p2)

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