The von Neumann entropy asymptotics in multidimensional fermionic systems

We study the von Neumann entropy asymptotics of pure translation-invariant quasi-free states of d-dimensional fermionic systems. It is shown that the entropic area law is violated by all these states: apart from the trivial cases, the entropy of a cubic subsystem with edge length L cannot grow slower than L^{d-1}ln L. As for the upper bound of the entropy asymptotics, the zero-entropy-density property of these pure states is the only limit: it is proven that arbitrary fast sub-L^d entropy growth is achievable.