7 -0.45 -0.3 -0.15  0 0  0.1  0.2  0.3  0.4 /U x = 2dt/U (a) Two dimensions (V=-0.85U) n=1 n=2 n=3 tp/th ext sp/sh ext third order -0.45 -0.3 -0.15  0  0  0.1  0.2  0.3  0.4 /U x = 2dt/U (b) Three dimensions (V=-0.85U) n=1 n=2 n=3 tp/th ext sp/sh ext third order d = 2 d = 3 V/U n = 1 n = 2 n = 1 n = 2 -0.01 0.0543 0.0337 0.0611 0.0379 -0.03 0.0937 0.0582 0.105 0.0655 -0.05 0.121 0.0749 0.136 0.0843 -0.07 0.142 0.0883 0.160 0.0994 -0.1 0.169 0.105 0.190 0.118 -0.2 0.233 0.145 0.262 0.164 -0.3 0.277 0.173 0.311 0.195 -0.4 0.307 0.193 0.345 0.217 -0.5 0.325 0.205 0.366 0.230 -0.6 0.331 0.209 0.372 0.235 -0.7 0.321 0.203 0.362 0.228 -0.8 0.291 0.183 0.327 0.206 -0.9 0.225 0.141 0.253 0.159 -0.93 0.193 0.121 0.217 0.136 -0.95 0.166 0.103 0.187 0.116 -0.97 0.1304 0.0812 0.147 0.0913 -0.99 0.0764 0.0474 0.0860 0.0534 FIG. 2. (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 . 85 U , n ↑ = n ↓ = n , and µ ↑ = µ ↓ = µ . The dotted lines correspond to phase boundary for the Mott insulator to superfluid state determined from the third-order strong-coupling expansion, the hollow pink-squares to the extrapolation fit for the single- particle or single-hole excitations (shown only for illustration purposes), and the solid black-circles to the extrapolation fit for the two- particle or two-hole excitations (the expected transition) discussed in the text. that the two-hole excitation branch has a negative slope in ( µ$_{↑}$ + µ$_{↓}$ ) / 2 versus t$_{σ}$ phase diagram when t$_{σ}$ → 0 ], i.e. − (2 n$_{↑}$n$_{↓}$/U$_{↑↓}$ ) zt$_{↑}$t$_{↓}$ − ∑ $_{σ}$[ n 2 $_{σ}$/U$_{↑↓}$ − ( n 2 σ − 1) / (2 U$_{σσ}$ + U$_{↑↓}$ ) + 2 n$_{σ}$ ( n$_{σ}$ + 1) /U$_{σσ}$ ] zt 2 σ term, which occurs for the first few Mott lobes beyond a critical U$_{↑↓}$ . When this coefficient is negative, its value is most negative for the first Mott lobe , We analyzed the zero temperature phase diagram of the two-species Bose-Hubbard (BH) model with on-site boson- boson interactions in d -dimensional hypercubic lattices, in- V. CONCLUSIONS What happens when t$_{↑}$ = t$_{↓}$ and/or U$_{↑↑}$ = U$_{↓↓}$ ? We donot expect any qualitative change for attractive interspecies inter- actions. However, for repulsive interspecies interactions, this lifts the degeneracy of the single-particle or single-hole exci- tation energies. While the transition is from a double Mott insulator to a double superfluid of both species in the degen- erate case, it is from a double-Mott insulator of both species to a Mott insulator of one species and a superfluid of the other in the nondegenerate case. TABLE II. List of the critical points (location of the tips) x c = 2 dt c /U that are found from the chemical potential extrapolation technique described in the text. Here, t $_{↑}$= t $_{↓}$= t , U $_{↑↑}$= U $_{↓↓}$= U , U ↑↓ = V , n ↑ = n ↓ = n , and µ ↑ = µ ↓ = µ . These critical points for the two-particle or two-hole excitations are determined from Eqs. (11) and (12) when V < 0 . Note that, for small V , x c ’s tend to increase as a function of V , since the presence of a nonzero V is what allowed these states to form in the first place. Howeve r, x c ’s decrease beyond a critical V , and they eventually vanish when V = − U , which may indicate an instability towards a collapse. and therefore the effect is strongest there. However, the coef- ficient increases and eventually becomes positive as a funct ion of filling, and thus the re-entrant behavior becomes weaker a s filling increases, and it eventually disappears beyond a cri tical filling. For the parameters used in Fig. 2, this occurs only fo r the first lobe, as can be seen in the figures. We also note that the sign of this coefficient is independent of the dimensiona l- ity of the lattice, since z = 2 d enters into the coefficient only as an overall factor.