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: 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?


1 Answer 1


Nice question. Luckily the answer turns out to be pretty simple! Your intuition is correct: the massless KG field has a logarithmic divergence in system size, and the massive case does not. The reason that both have divergences in that article is because they are doing QFT and hence taking two limits: infinite system size, and zero UV cut-off. The massless case has divergences due to both limits, but the massive case only due to the latter. Often the area law is phrased in the context of many-body systems/condensed matter, where one usually does not take the latter limit, and in that case massive systems have no divergences -- consistent with your understanding.

To be a bit more exact, in that article they essentially prove two different formulas in the case of $D = 2 \; (=1+1)$. To summarize the notation: $L$ is (half) the system size, $S$ is the entanglement entropy from cutting it in two, $\epsilon$ is the UV cut-off, and $\mu$ is the mass of the KG field.

  1. On page 10 they prove $$S = \frac{1}{6} \ln \left( \frac{L}{\epsilon} \right) + \mathcal O \left(L^0\right) \qquad \left(\textrm{for } \mu = 0 \right)$$
  2. On page 12 they prove (where I correct a typo) $$S = \frac{1}{6} \ln \left( \frac{1}{\mu\epsilon} \right) \qquad \qquad \quad \left(\textrm{for } L = \infty \right)$$

You see that for a fixed $\epsilon$ (as we might do when we are actually looking at lattice systems such as in condensed matter), we get the usual result: (1) the massless case has a logarithmic divergence in $L$, whereas (2) the massive case does not.

Note that if we take the UV cut-off to zero, the massless case has two divergences! One due to $L\to \infty$, the other due to $\epsilon \to 0$. Also, the first formula is consistent with the intuition that a massless field in a finite box acquires a zero-point mass $\mu_\textrm{eff} \sim \frac{1}{L}$, consistent with (2).


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