# Can the entanglement entropy of a ground state scale extensively?

To my knowledge, entanglement entropy (EE) for the ground state of a many-body system scales subextensively, that is: $$\lim_{\text{Vol}_A \to \infty} \frac{S_A}{\text{Vol}_A}\to 0.$$

Here, I have in mind a bipartition $A\cup \bar{A}$ of a ground state $\rho = |\psi><\psi|$ for some physical Hamiltonian (e.g. think of spin systems); $\rho_A = \text{Tr}_{\bar{A}}\rho$ is the reduced density matrix for the partition $A$ and $\text{Vol}_A = \ell^{d}$ is its volume (e.g. in 1D it is the length $\ell$).

Now, as far as I know, possible behaviour of $S_A$ changing the scales are:

1. Area Law: $S_A \sim \text{Area}(\partial A)\sim \ell^{d-1}$ (and I think this is believed for gapped system)
2. Log corrections: $S_A \sim \ell^{d-1} log l$ (which is proved for some specific model and 1+1 CFT)
3. Power law: $S_A \sim \ell^\alpha$ with $\alpha < d$. This latter is actually proved just for 1d system by Schor quite recently (paper: 1408.1657).

Anyway, it seems that in any case $S_A$ cannot scale as $\text{Vol}_A$; my question is: is there any proof of this?