  While volume violation of area law has been exhibited in several quantum spin
chains, the construction of a corresponding model in higher dimensions, with
isotropic terms, has been an open problem. Here we construct a 2D
frustration-free Hamiltonian with maximal violation of the area law. We do so
by building a quantum model of random surfaces with color degree of freedom
that can be viewed as a collection of colored Dyck paths. The Hamiltonian may
be viewed as a 2D generalization of the Fredkin spin chain. Its action is shown
to be ergodic within the Hilbert subspace of zero fixed Dirichlet boundary
condition and positive height function in the bulk and exhibits a
non-degenerate ground state. Its entanglement entropy between subsystems
exhibits an entanglement phase transition as the deformation parameter is
tuned. The area- and volume-law phases are similar to the one-dimensional
model, while the critical point scales with the linear size of the system $L$
as $L\log L$. Similar models can be built in higher dimensions with even softer
area law violations at the critical point.
