  In a bipartite set-up, the vacuum state of a free Bosonic scalar field is
entangled in real space and satisfies the area-law--- entanglement entropy
scales linearly with area of the boundary between the two partitions. In this
work, we show that the area law is violated in two spatial dimensional model
Hamiltonian having dynamical critical exponent z=3. The model physically
corresponds to next-to-next-to-next nearest neighbour coupling terms on a
lattice. The result reported here is the first of its kind of violation of area
law in Bosonic systems in higher dimensions and signals the evidence of a
quantum phase transition. We provide evidence for quantum phase transition both
numerically and analytically using quantum Information tools like entanglement
spectra, quantum fidelity, and gap in the energy spectra. We identify the cause
for this transition due to the accumulation of large number of angular zero
modes around the critical point which catalyses the change in the ground state
wave function due to the next-to-next-to-next nearest neighbor coupling.
Lastly, using Hubbard-Stratanovich transformation, we show that the effective
Bosonic Hamiltonian can be obtained from an interacting fermionic theory and
provide possible implications for condensed matter systems.
