and this is discernible in these figures.

$V_{12}>0$
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One of the phenomena unique to TBEC is the phase separation. This provides important insights to understand novel phenomena in nonlinear dynamics, pattern formation, quantum phase transitions in condensed matter systems, etc. [@gautam_10_1; @gautam_10_2; @gautam_11; @roy_15_2; @bandyopadhyay_17; @gautam_12; @gautam_13; @roy_14_2; @kuopanportti_19; @roy_14_1; @suthar_15; @roy_15_1; @suthar_16; @suthar_17; @pal_17; @pal_18] Phase separation of TBECs in the weakly interacting regime, as mentioned earlier, is well studied. This, however, is not the case for the strongly interacting two-species ultracold atoms in optical lattices.

![Phase diagram of extended BHM for TBEC at the different inter-species interaction strength $U_{12}$ and for inter-species NN interaction $V_{12} = V_{21} = 0.05$, $V_1 = V_2 = 0.05U$. The incompressible (MI, cDW) and compressible phase(SS, SF) regions are separated by brown lines. In cDW phase the two species occupy lattice sites randomly in such a way such that total density $\rho= \rho^1 +\rho^2$ have DW pattern. Around DW phase, SS phase exists and the boundary of the SS region is marked by green lines. (d) For $U_{12} = 1.2$ the DW and SF phases are phase separated.[]{data-label="ebhm_v12_0p05"}](ebh_v12p0p05_ph.pdf){width="8.0cm"}

As discussed earlier, in the TBHM we observe phase separation in the SF phase, 