  We examine the quantum Hall (QH) states of the optical lattices with square
geometry using Bose-Hubbard model (BHM) in presence of artificial gauge field.
In particular, we focus on the QH states for the flux value of $\alpha = 1/3$.
For this, we use cluster Gutzwiller mean-field (CGMF) theory with cluster sizes
of $3\times 2$ and $3\times 3$. We obtain QH states at fillings $\nu = 1/2, 1,
3/2, 2, 5/2$ with the cluster size $3\times 2$ and $\nu = 1/3, 2/3, 1, 4/3,
5/3, 2, 7/3, 8/3$ with $3\times 3$ cluster. Our results show that the geometry
of the QH states are sensitive to the cluster sizes. For all the values of
$\nu$, the competing superfluid (SF) state is the ground state and QH state is
the metastable state.
