order parameters at lattice sites with $(p,q)$ belonging to sublattices A and B, respectively. The compressibility of each species are calculated using the definition $\partial \mu^k/\partial \rho^k$.

The TBHM, like the single species BHM, shows two phases MI and SF phases. The MI phase is an incompressible phase with integer commensurate density $n^{k,A}=n^{k,B}\in\mathbb{N}$. And, incompressibility implies zero SF order parameter $\phi^{k,A}=\phi^{k,B}=0$. The SF phase, on the other hand is compressible. Hence, it has $n^{k,A}=n^{k,B}\in \mathbb{R}$, $\phi^{k,A}=\phi^{k,B}\in \mathbb{R}$ and $\kappa^k\neq0$. For these two phases, the density and SF order parameters are uniform, so the contrast order parameters $\Delta\rho^k$ and $\Delta\phi^k$ are zero. In the eTBHM, the NN interaction leads to the emergence of two more quantum phases, DW and SS. These two phases have non-uniform density and SF order parameters. As a result the distinguishing features of these phases are non-zero contrast order parameters. The DW phase has integer $n^k_{p,q}$ with $n^{k,A}\neq n^{k,B}$ and $\Delta \rho^k\in\mathbb{N}$. This phase has zero SF order parameter $\phi^{k,A}=\phi^{k,B}=0$ and hence, incompressible. The SS phase has real $n^k_{p,q}$ with $n^{k,A}\neq n^{k,B}$ and $\Delta \rho^k\in\mathbb{R}$. The SF order parameter in this phase is non-zero and non-uniform. This implies that $n^{k,A}\neq n^{k,B}$ and $\phi^{k,A}\neq\phi^{k,B}$. So, both the contrast 