phase diagram of eTBHM. In particular, the miscible and immiscible phases. We end the paper with conclusion in Section \[sec\_conclude\].

Theory {#sec_theory}
======

TBHM Hamiltonian
----------------

At zero temperature, the TBHM Hamiltonian which describes the physics of a TBEC in 2D optical lattice is [@damski_03] $$\begin{aligned}
\hat{H}^{{\rm TBH}} &=& -\sum_{p, q, k}\bigg [ \Big( J_x^k 
              \hat{b}_{p+1, q}^{\dagger k}\hat{b}_{p, q}^k + {\rm H.c.}\Big)
              + \Big( J_y^k \hat{b}_{p, q+1}^{\dagger k}
              \hat{b}_{p, q}^k  \nonumber\\ 
              &&+ {\rm H.c.}\Big) - \frac{U_{kk}}{2}
              \hat{n}_{p, q}^k (\hat{n}_{p, q}^k-1) + 
              \tilde{\mu}^k_{p,q}\hat{n}_{p, q}^k\bigg]
              \nonumber\\
              &&+\sum_{p, q} U_{12}\hat{n}_{p, q}^1 \hat{n}_{p, q}^2,
\label{tbhm}  \end{aligned}$$ where $k = 1$,$2$ is the species index, $(p,q)$ are the lattice indices, $J_x^k$ ($J_y^k$) is the nearest neighbour (NN) hopping strength along $x$ ($y$) directions, $\hat{b}^{\dagger k}_{p,q}$ ($\hat{b}^k_{p,q}$) is the creation (annihilation) operator and $\hat{n}_{p, q}^k$ is the number operator at ($p,q$)th site. $U_{kk}$ is intra-species interaction strength, and $U_{12}$ is inter-species interaction strength between two species. Further, $\tilde{\mu}^k_{p,q} =\mu^k - \varepsilon_{p,q}^k$, is the local chemical potential at each site for the two species where $\varepsilon_{p,q}^k$ is envelop potential for the species. For a system of $K \times L$ lattices sites, the index along $x$ ($y$) has values $p=1,\ldots K$ ($q=1,\ldots L$). The unique feature of the 