  The sub-volume scaling of the entanglement entropy with the system's size,
$n$, has been a subject of vigorous study in the last decade [1]. The area law
provably holds for gapped one dimensional systems [2] and it was believed to be
violated by at most a factor of $\log\left(n\right)$ in physically reasonable
models such as critical systems.
  In this paper, we generalize the spin$-1$ model of Bravyi et al [3] to all
integer spin-$s$ chains, whereby we introduce a class of exactly solvable
models that are physical and exhibit signatures of criticality, yet violate the
area law by a power law. The proposed Hamiltonian is local and translationally
invariant in the bulk. We prove that it is frustration free and has a unique
ground state. Moreover, we prove that the energy gap scales as $n^{-c}$, where
using the theory of Brownian excursions, we prove $c\ge2$. This rules out the
possibility of these models being described by a conformal field theory. We
analytically show that the Schmidt rank grows exponentially with $n$ and that
the half-chain entanglement entropy to the leading order scales as $\sqrt{n}$
(Eq. 16). Geometrically, the ground state is seen as a uniform superposition of
all $s-$colored Motzkin walks. Lastly, we introduce an external field which
allows us to remove the boundary terms yet retain the desired properties of the
model. Our techniques for obtaining the asymptotic form of the entanglement
entropy, the gap upper bound and the self-contained expositions of the
combinatorial techniques, more akin to lattice paths, may be of independent
interest.
