for the MI phase with $\rho=1$. The density pattern has one atom at each lattice site chosen randomly from the two species. Thus, at a given lattice site $(p,q)$ we can have the occupancies as $n^{1}_{p,q} =1, n^{2}_{p,q} =0$ or $n^{1}_{p,q} =0, n^{2}_{p,q} =1$. In the perturbative analysis, without loss of generality, we consider neighbouring lattice sites are occupied by atoms of different species. This is also one realization of the energetically favourable configuration for $U_{12}<U$. Then, with the correction arising from $b^{\dagger 1}\phi^{1}$, the equation $$\begin{aligned}
   \frac{1}{4J}= \frac{2}{U_{12} -\mu} +\,\frac{1}{\mu},
   \label{tbhm_odd}\end{aligned}$$ defines the phase boundary of the MI lobe with $\rho=1$. Based on similar analysis, we can obtain the phase boundary of other MI lobes. For which we have to use Eq.(\[tbhm\_even\]) and (\[tbhm\_odd\]) for the even and odd integer values of $\rho$, respectively.

### Extended TBHM

The MI-SF boundary of the eTBHM can also be calculated similar to the TBHM. The expression of the order parameter is similar to Eq. (\[order\_par\_ana\]) but $\bar{\mu}^{k}$ is given by $$\begin{aligned}
   \bar{\mu}^{k}_{p,q} &= &\tilde{\mu}^{k}_{p,q}-U_{12}n^{3-k}_{p,q}
        -4V_k n^{k}_{p,q} -4V_{12} n^{3-k}_{p,q},\end{aligned}$$ For MI phase with $\rho=2n$, the occupancies are $n^{k}_{p,q}=n$. Further, assuming $V_1 =V_2$, the MI-SF boundary is given by $$\begin{aligned}
   \frac{1}{4J}= \frac{2}{U -\bar{\mu}} +\,\frac{1}{\bar{\mu}}.
 