\label{exTBHM_MI2}\end{aligned}$$ with $\bar{\mu} = \mu -U_{12} -4V_1 -4V_{12}$. A similar analysis of MI phase with odd integer occupancies require specific approximations based on the density configuration. And, a general unified analysis in not applicable.

Characterization of the phases
------------------------------

  --------------------- -------- ---------- ---------------- ----------------
                        $\rho$   $\phi$     $\Delta\rho^k$   $\Delta\phi^k$
  Mott Insulator (MI)            $ = 0$     $ = 0$           $ = 0$
  Superfluid (SF)                $ \ne 0$   $ = 0$           $ = 0$
  Density Wave (DW)              $ = 0$     $ \ne 0$         $ = 0$
  Super Solid (SS)               $ \ne 0$   $ \ne 0$         $ \ne 0$
  --------------------- -------- ---------- ---------------- ----------------

  : Classification of different quantum phases with order parameters for our systems.[]{data-label="table"}

To identify different quantum phases of the system we compute the density contrast $\Delta\rho^k$, order parameter contrast $\Delta\phi^k$ and compressibility $\kappa^k$. To define $\Delta\rho^k$, divide the lattice site occupancies as $$n^k_{p,q}=\begin{cases}
                n^{k,A} & \text{if $(p,q) \in$ sublattice A}, \\
                n^{k,B} & \text{if $(p,q) \in$ sublattice B},
             \end{cases}
   \label{occ_def}$$ then, the density contrast of the $k$th species is $$\Delta\rho^k = n^{k,A}-n^{k,B}.
   \label{den_con}$$ The order parameter contrast is defined similarly as $$\Delta\phi^k = \phi^{k,A}-\phi^{k,B},
   \label{den_con}$$ where, $\phi^{k,A}$ and $\phi^{k,B}$, like in the case of density are the values of the 