Entanglement entropy in (1+1)d field theory with dynamical critical exponent $z>1$

Question

It was well known that for (1+1)d CFT(z=1) case, we can use the tool of conformal map to derive the formula of entanglement entropy for a finite interval: S ~ $c \log L$. L is the length of the interval.(allow me to be sloppy, i neglect the coefficient and cutoff in the formula).

My question is: is there a derivation in literature or in your mind that shows the entanglement entropy formula for other dynamical critical exponent? for example z=2(quadratic dispersion) or z=3, in a (1+1)d field theory model.

I'd like to know how the formula looks like, $\log L$ or $L^{\alpha}$ for some exponent $\alpha$ determined by the universality class of the theory.

Answer 1

For $z=2$ there is a study here http://arxiv.org/abs/quant-ph/0404026 in the context of ferromagnetic spin chain. The result is basically $\log L$. Swingle and Senthil argued in http://arxiv.org/abs/1112.1069 that "generally" the violation of area law for EE is at most $L^{d-1}\log L$ where $d$ is the space dimension. However, http://arxiv.org/abs/1408.1657 constructed a frustration-free translation-invariant gapless spin chain with power-law divergence of EE (they prove an upper bound for spectral gap as $O(L^{-c}$ where $c\geq 2$ and $L$ is the system size, so this is not a CFT). I'm not sure how general the latter work is, or just some special "pathological" example.