parameters, the ground state of the next lattice site is computed and this process is repeated till all the lattices sites are covered. One such sweep is identified as an iteration and we then, start the process again for the next iteration. The iterations are carried out till the convergence criterion $|\phi_{p,q}^{n-1} - \phi_{p,q}^n| \lesssim 10^{-12}$ is satisfied at the $n^{{\rm th}}$ iteration. In the present work, to determine the phase diagrams, we consider lattice system of size $10 \times 10$ and choose $N_b = 7$. That is, $K$ and $L$ are both 10. We find that the phase boundaries remain unchanged when the system size is augmented to $20\times 20$. We also use the augmented system size to validate key findings. In addition, we employ periodic boundary conditions to model an infinite-sized system.

Extended TBHM
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The BHM with NN interaction, referred to as the extended BHM (eBHM), exhibits richer phase diagram than BHM and it has the novel feature of harbouring the SS phase. The phase diagram of the single species eBHM consists of DW, SS, MI and SF phases. Similarly, the extended TBHM (eTBHM) also exhibits these phases as well as miscible and segregated phases and the 