been some works on 2D as well [@altman_03; @chen_03; @kuklov_03; @kuklov_04; @isacsson_05; @iskin_10; @anufriiev_16]. The phase diagram of TBHM shows different combinations of mixed MI-SF phases apart from the MI and SF phases. And, these have been investigated using quantum Monte-Carlo [@kuklov_03; @kuklov_04], mapping to spin systems [@altman_03], and with mean field theory [@chen_03; @isacsson_05; @iskin_10; @pai_12; @anufriiev_16]. These studies, except for Ref. [@pai_12], considered homogeneous systems. However, hitherto the phenomenon of phase separation in 2D TBHM is yet to be investigated in detail.

The quantum phases of TBHM in the phase separated domain, unlike in the TBECs, do not show segregation into two spatial domains. We attribute this to the lack of long range interaction in the TBHM. The simplest modification to include the effect of long range interaction is to add nearest neighbour interaction. The eBHM, as mentioned earlier, supports two more quantum phases density wave (DW) [@capogrosso_10; @flottat_17; @iskin_11] and supersolid (SS) phase [@ng_08; @iskin_11; @yamamoto_11; @boninsegni_12; @leonard_17]. The DW phase is an insulating phase similar to the MI phase but it has crystalline order or diagonal long range order. And, the SS phase is a compressible phase with both diagonal and off diagonal long range order. 