TBECs is the phase separation and for continuum systems, the criterion for phase segregation is $U_{12}^2 > U_{11}U_{22}$ [@ho_96; @Trippenbach_00]. Otherwise, the TBEC is in the miscible phase. For the case of strongly interacting TBECs in optical lattices, described by the above Hamiltonian, we show the existence of different phases in both the miscible and immiscible domains.

To obtain the ground state of the TBHM Hamiltonian in Eq. (\[tbhm\]), we use single site Gutzwiller mean field (SGMF) theory [@rokhsar_91; @sheshadri_93; @bai_18; @pal_19; @bandyopadhyay_19; @kuldeep_19]. The starting point of this theory is to separate the operators into mean-field and fluctuation operator components as $\hat{b}_{p, q}^k = \phi_{p,q}^k + \delta \hat{b}_{p, q}^k\;$ and $\;\hat{b}^{\dagger k}_{p,q} = \phi_{p,q}^{k*} + 
\delta\hat{b}_{p,q}^{\dagger k}$. Then, the Hamiltonian in Eq. (\[tbhm\]) is reduced to sum of single-site mean-field Hamiltonian $$\begin{aligned}
\hat{h}_{p,q}^{{\rm TBH}} &= &- \sum_{k}\left[J_x^k 
                  \left(\hat{b}_{p+1, q}^{\dagger k}\phi_{p,q}^k 
                  + \phi^{k*}_{p + 1, q}\hat{b}_{p, q}^k \right ) + {\rm H.c.}
                  \right.  \nonumber\\
                  && +  J_y^k\left(\hat{b}_{p, q+1}^{\dagger k} \phi_{p,q}^k 
                  +  \phi^{k*}_{p, q+1}\hat{b}_{p, q}^k  \right) + {\rm H.c.}
                  \nonumber \\
                  &&-\left . \frac{U_{kk}}{2}\hat{n}_{p, q}^k
                  \left (\hat{n}_{p, q}^k-1\right) 
                  + \tilde{\mu}^k_{p,q}\hat{n}_{p, q}^k\right]
                  \nonumber\\
                  &&+ U_{12}\hat{n}_{p, q}^1 \hat{n}_{p, q}^2,
\label{ham_ss_tbec}\end{aligned}$$ where, $\phi_{p,q}^k$ ($\phi_{p,q}^{k*}$) is the SF order parameter. With this definition of the 