different values of $U_{12}$ are shown in Fig. \[tbec\_ph\].

For simplicity, we consider symmetric hopping $J_x^k = J_y^k = J$, equal chemical potential $\tilde{\mu}^1_{p,q} = \tilde{\mu}^2_{p,q} = \mu$ and identical intra-species interactions $U_{kk} = U$. We scale all the energies with $U$, and define the phase diagram in the $J/U-\mu/U$ plane. The phase diagram consists of a sequence of MI lobes having integer $\rho$. Without the inter-species interaction $U_{12} = 0$, as shown in Fig. \[tbec\_ph\](a), the phase diagram is equivalent to the case of single species. But, with twice the occupancy. That is the MI lobes have $\rho=2n$ with $\rho^1=\rho^2=n$ and $n\in\mathbb{N}$. So, the lowest MI lobe has $\rho=2$ and each lattice has one atom from each of the two species. As a result, the phase diagram is identical to single species. With the introduction of the inter-species interaction ($U_{12} \neq 0$) the half filled lobes like $\rho = 1$ emerge in the phase diagram with $0 < \rho^1 <1$, and then, $\rho^2 = 1-\rho^1$. This is discernible for $U_{12} =0.4 U$ from the Fig. \[tbec\_ph\](b). Based on the form of the interactions in the Hamiltonian of the system, the energies of system is degenerate for all the possible 