We generalize the area-law violating models of Fredkin spin chain into two
dimensions by building a quantum bicolor six-vertex model with correlated
swapping and alternating boundary condition. The construction is a analogous to
our first example of such models based on quantum lozenge tiling
(arXiv:2210.01098), but on a square lattice. The Hamiltonian is frustration
free, and its projectors generate ergodic dynamics within the subspace of
height configuration that are non negative. The ground state is a volume- and
color-weighted superposition of classical bicolor six-vertex configurations
with non-negative heights in the bulk an zero height on the boundary. The
entanglement entropy between subsystems has a phase transition as the weight
parameter is tuned, which is shown to be robust in the presence of an external
field acting on the color degree of freedom. The ground state transitions
between area- and volume-law entanglement phases with a critical point where
entanglement entropy scales as a function $L\log L$ of the linear system size
$L$. Intermediate scalings between $L\log L$ and $L^2$ can be achieved with an
inhomogeneous deformation parameter that approaches 1 at different rates in the
thermodynamic limit.
