  We show that, prepotential formulation of gauge theories on honeycomb lattice
yields local loop states, which are free from any spurious loop degrees of
freedom and hence exact and orthonormal. We also illustrate that, the dynamics
of orthonormal loop states are exactly same in both the square and honeycomb
lattices. We further extend this construction to arbitrary dimensions.
Utilizing this result, we make a mean field ansatz for loop configurations for
SU(2) lattice gauge theory in $2+1$ dimension contributing to the low energy
sector of the spectrum. Using variational analysis, we show that, this type of
mean loop configurations has two distinct phases in the strong and weak
coupling regime and shows a first order transition at $g=1$. We then propose a
reduced Hamiltonian to describe the dynamics of the theory within the mean
field ansatz. We further work with the mean loop configuration obtained at the
weak coupling regime and analytically calculate the spectrum of the reduced
Hamiltonian. The spectrum matches with that of the existing literature in this
regime, establishing our ansatz to be a valid alternate one which is far more
easier to handle for computation.
