phase separation criterion is met, there is no change in the phase diagram with further increase in $U_{12}$. As an example the phase diagram for $U_{12}=1.2U$ is shown in Fig.\[tbec\_ph\](d). The lobes in this phase diagram are the same as in Fig.\[tbec\_ph\](a). The only difference is the occupancy is $\rho=n$ with $n\in\mathbb{N}$ and $\rho^k=n/2$. As a result, the density pattern of the lowest MI lobe ($\rho=1$) has one atom at each lattice site chosen randomly from the two species. The important point is, the MI lobes have the same sizes for $U_{12}=0$ and $U_{12}\geqslant U_{kk}$. But, the occupancy and hence the density patterns are different.

Phase diagram with long range interactions {#sec_phdiag_ext}
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$V_{12}=0$
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![Phase diagram of extended BHM for TBEC at the different inter-species interaction strength $U_{12}$ and for inter-species NN interaction $V_{12} = V_{21} = 0$, $V_1 = V_2 = 0.05U$. Brown lines forms the boundary of region comprising of incompressible phases (MI, DW). Around DW phase, SS phase exists and the boundary between SS and SF phases is represented by green lines. The SS region around the DW region enlarges with increasing $U_{12}$. In DW(n,0) phase both species have DW(n,0) pattern and in MI(1,1) phase both 