6  0  1.5  3  4.5 0  0.09  0.18  0.27 /U x = 2dt/U (a) Two dimensions (V=0.5U) n=1 n=2 n=3 sp/sh ext third order  0  1.5  3  4.5  0  0.09  0.18  0.27 /U x = 2dt/U (b) Three dimensions (V=0.5U) n=1 n=2 n=3 sp/sh ext third order disappears at a critical value of t , beyond which the system becomes a superfluid. In addition, similar to what was found for the single-species BH model [19, 24], the strong-coupling expansion overestimates the phase boundaries, and it leads to unphysical pointed tips for all Mott lobes, which is expected since a finite-order expansion cannot describe the physics o f the critical point correctly. A short list of V/U versus the crit- ical points x$_{c}$ = 2 dt$_{c}$/U is presented for the first two Mott insulator lobes in Table I, where it is shown that the criti- cal points tend to move in as V increases. This is because presence of a second species (say − σ ones) screens the on- site intraspecies repulsion U$_{σσ}$ between σ -species, and hence increases the superfluid region. In Fig. 2, we show the chemical potential µ (in units of U ) versus x = 2 dt/U phase diagram for (a) two-dimensional and (b) three-dimensional hypercubic lattices, where in these figures we choose the interspecies interaction to be attract ive V = − 0 . 85 U . Comparing Eqs. (8) and (9) with Eqs. (11) and (12), we expect that the excited state of the system to be a paired superfluid for all V < 0 when t → 0 . This is clearly seen in the figure where the dotted lines correspond t o phase boundary for the Mott insulator to superfluid state as d e- termined from the third-order strong-coupling expansion, the hollow pink-squares correspond to the extrapolation fits fo r the single-particle and single-hole excitations (shown only for illustration purposes), and the solid black-circles correspond to the extrapolation fits for the two-particle and two-hole e x- citations (this is the expected transition) discussed in the text. At t = 0 , the chemical potential width of all Mott lobes are V + U = 0 . 15 U , which is in contrast with the single- species BH model. As t increases from zero, the range of µ about which the ground state is a Mott insulator decreases here as well, and the Mott insulator phase disappears at a critical value of t , beyond which the system becomes a paired super- fluid. The strong-coupling expansion again overestimates t he phase boundaries, and it again leads to unphysical pointed tips for all Mott lobes. In addition, a short list of V/U versus the critical points x$_{c}$ = 2 dt$_{c}$/U are presented for the first two Mott insulator lobes in Table I. Our results are consistent with the expectation that, for small V , the locations of the tips in- crease as a function of V , because the presence of a nonzero V is what allowed these states to form in the first place. How- ever, when V is larger than some critical value ( ∼ 0 . 6 U ), the locations of the tips decrease, and they eventually vanish when V = − U . This may indicate an instability towards a collapse since at this point U$_{↑↑}$U$_{↓↓}$ is exactly equal to U 2 $_{↑↓}$. FIG. 1. (Color online) Chemical potential µ (in units of U ) versus x = 2 dt/U phase diagram for (a) two- and (b) three-dimensional hypercubic lattices with t ↑ = t ↓ = t , U ↑↑ = U ↓↓ = U , U ↑↓ = V = 0 . 5 U , n ↑ = n ↓ = n , and µ ↑ = µ ↓ = µ . The dotted lines correspond to phase boundary for the Mott insulator to superfluid state as determined from the third-order strong-coupling expansion, and the hollow pink-squares to the extrapolation fit for the s ingle- particle or single-hole excitations discussed in the text. Recall that an incompressible super-counter flow phase also exists outs ide of the Mott insulator lobes. Compared to the V > 0 case shown in Fig. 1, note that shape of the Mott insulator to paired superfluid phase bound- ary is very different, showing a re-entrant behavior in all di- mensions from paired superfluid to Mott insulator and again to a paired superfluid phase, as a function of t . Our results are consistent with an early numerical time-evolving block decimation (TEBD) calculation [11], where such a re-entrant quantum phase transition in one dimension was predicted. The re-entrant quantum phase transition occurs when co- efficient of the hopping term in Eq. (12) is negative [so At t = 0 , the chemical potential width of all Mott lobes are U (similar to the single-species BH model), but they are separated from each other by V as a function of µ . As t in- creases from zero, the range of µ about which the ground state is a Mott insulator decreases, and the Mott insulator phase