5 Kosterlitz-Thouless shapes for d = 1 and power-law shapes for d > 1 . For illustration purposes, here we analyze only the latter case, but this technique can be easily adapted to the d = 1 case [19]. is known via other means, e.g. numerical simulations, α , β , γ and δ can be calculated accordingly, for which the extrap- olation technique gives very accurate results [23, 25]. If the exact value of x$_{c}$ is not known, then we set δ = 0 , and solve Eqs. (14), (15), (16) and the δ = 0 equation to determine α , β , γ and x$_{c}$ self-consistently, which also leads to accurate results [19, 24]. Next we present typical ground-state phase diagrams for ( d = 2) - and ( d = 3 )-dimensional hypercubic lattices obtained from this extrapolation technique. From now on we consider a two-species mixture with t$_{↑}$ = t$_{↓}$ = t , U$_{↑↑}$ = U$_{↓↓}$ = U , U$_{↑↓}$ = V , n$_{↑}$ = n$_{↓}$ = n , and µ$_{↑}$ = µ$_{↓}$ = µ . When d > 1 , we propose the following ansatz which includes the known power-law critical behavior of the tip of the lobes In Figs. 1 and 2, the results of the third-order strong- coupling expansion (dotted lines) are compared to those of the extrapolation technique (hollow pink-squares and solid black- circles) when V = 0 . 5 U and V = − 0 . 85 U , respectively, in two ( d = 2 or z = 4 ) and three ( d = 3 or z = 6 ) dimensions. We recall here that t$_{↑}$ = t$_{↓}$ = t , U$_{↑↑}$ = U$_{↓↓}$ = U , U$_{↑↓}$ = V , n$_{↑}$ = n$_{↓}$ = n , and µ$_{↑}$ = µ$_{↓}$ = µ . where A ( x ) = a + bx + cx $^{2}$+ dx $^{3}$+ · · · and B ( x ) = α + βx + γx $^{2}$+ δx $^{3}$+ · · · are regular functions of x = 2 dt/U , x$_{c}$ is the critical point which determines the location of the lobes, and zν is the critical exponent for the ( d + 1 )-dimensional XY model which determines the shape of the lobes near x$_{c}$ = 2 dt$_{c}$/U . In Eq. (13), the plus sign corresponds to the particle branch, and the minus sign corresponds to the hole branch. The form of the ansatz is taken to be the same for both single- and two-partice (or single- and two-hole) excitations, but the parameters are very different. B. Numerical Results A. Scaling Ansatz µ ± U = A ( x ) ± B ( x )( x$_{c}$ − x ) $^{zν}$, (13) In Fig. 1, 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 we choose the interspecies interaction to be repulsive V = 0 . 5 U . Com- paring Eqs. (8) and (9) with Eqs. (11) and (12), we expect that the excited state of the system to be the usual superfluid for all V > 0 for all t . The dotted lines correspond to phase boundary for the Mott insulator to superfluid state as deter- mined from the third-order strong-coupling expansion, and the hollow pink-squares correspond to the extrapolation fit s for the single-particle and single-hole excitations discussed in the text. We recall here that an incompressible super-counter flow phase [7–9, 13] also exists outside of the Mott insulator lobes, but our current formalism cannot be used to locate its phase boundary. The parameters a , b , c and d depend on U , V and n , and they are determined by matching them with the coefficients given by our third-order expansion such that A ( x ) = ( µ $^{par}$+ µ $^{hol}$) / (2 U ) . Here, µ par and µ hol are our strong-coupling ex- pansion results determined from Eqs. (8) and (9) for the single-particle and single-hole excitations, or from Eqs. (11) and (12) for the two-particle and two-hole excitations, respec- tively. Writing our strong-coupling expansion results for the particle and hole branches in the form µ par = U ∑ 3 n $_{=0}$e + $_{n}$x n and µ hol = U ∑ 3 n $_{=0}$e − $_{n}$x $^{n}$, leads to a = ( e + 0 + e − $_{0}$) / 2 , b = ( e + 1 + e − $_{1}$) / 2 , c = ( e + 2 + e − $_{2}$) / 2 , and d = ( e + 3 + e − $_{3}$) / 2 . To determine the U , V and n dependence of the parameters α , β , γ , δ , x$_{c}$ and zν , we first expand the left hand side of B ( x )( x$_{c}$ − x ) zν = ( µ par − µ $^{hol}$) / (2 U ) in powers of x , and match the coefficients with the coefficients given by our thir d- order expansion, leading to α = e + 0 − e − 0 2 x zν c , (14) β α = zν x$_{c}$ + e + 1 − e − 1 e + 0 e − 0 , (15) γ α = zν ( zν + 1) 2 x 2 c + zν x$_{c}$ e + 1 − e − 1 e + 0 e − 0 + e + 2 − e − 2 e + 0 e − 0 , (16) δ α = zν ( zν + 1)( zν + 2) 6 x 3 c + zν ( zν + 1) 2 x 2 c e + 1 − e − 1 e + 0 e − 0 + zν x$_{c}$ e + 2 − e − 2 e + 0 e − 0 + e + 3 − e − 3 e + 0 e − 0 . (17) d = 2 d = 3 V/U n = 1 n = 2 n = 1 n = 2 0.0 0.234 0.138 0.196 0.116 0.1 0.234 0.138 0.196 0.115 0.2 0.233 0.137 0.195 0.115 0.3 0.230 0.136 0.194 0.114 0.4 0.227 0.134 0.193 0.113 0.5 0.223 0.131 0.190 0.112 0.6 0.217 0.128 0.187 0.110 0.7 0.208 0.123 0.182 0.107 0.8 0.197 0.116 0.174 0.102 0.9 0.193 0.113 0.163 0.095 TABLE I. List of the critical points (location of the tips) x c = 2 dt c /U for the first two Mott insulator lobes 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 single-particle or single- hole excitations are determined from Eqs. (8) and (9), and they tend to move in as V increases, and are independent of the sign of V . We fix zν at its well-known values such that zν ≈ 2 / 3 for d = 2 and zν = 1 / 2 for d > 2 . If the exact value of x$_{c}$