Strong-coupling expansion for the two-species Bose-Hubbard model M. Iskin Department of Physics, Koc¸ University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey (Dated: August 28, 2018) To analyze the ground-state phase diagram of Bose-Bose mixtures loaded into d -dimensional hypercubic op- tical lattices, we perform a strong-coupling power-series expansion in the kinetic energy term (plus a scaling analysis) for the two-species Bose-Hubbard model with onsite boson-boson interactions. We consider both repulsive and attractive interspecies interaction, and obtain an analytical expression for the phase boundary be- tween the incompressible Mott insulator and the compressible superfluid phase up to third order in the hoppings. In particular, we find a re-entrant quantum phase transitionfrom paired superfluid (superfluidity of composite bosons, i.e. Bose-Bose pairs) to Mott insulator and again to a paired superfluid in all one, two and three di- mensions, when the interspecies interaction is sufficientl y large and attractive. We hope that some of our results could be tested with ultracold atomic systems. PACS numbers: 03.75.-b, 37.10.Jk, 67.85.-d Single-species Bose-Hubbard (BH) model is the bosonic generalization of the Hubbard model, and was introduced originally to describe $^{4}$He in porous media or disordered gran- ular superconductors [1]. For hypercubic lattices in all di- mensions d , there are only two phases in this model: an in- compressible Mott insulator at commensurate (integer) fill - ings and a compressible superfluid phase otherwise. The su- perfluid phase is well described by weak-coupling theories, but the insulating phase is a strong-coupling phenomenon that only appears when the system is on a lattice. Transition from the Mott insulator to the superfluid phase occurs as the hop- ping, particle-particle interaction, or the chemical potential is varied [1]. In this paper, we examine the ground-state phase diagram of the two-species BH model with on-site boson-boson interac- tions in d -dimensional hypercubic lattices, including both the repulsive and attractive interspecies interaction, via a strong- coupling perturbation theory in the hopping. We carry the expansion out to third-order in the hopping, and perform a scaling analysis using the known critical behavior at the tip of the insulating lobes, which allows us to accurately predict the critical point, and the shape of the insulating lobes in the plane of the chemical potential and the hopping. This tech- nique was previously used to discuss the phase diagram of the single-species BH model [19–23], extended BH model [24], and of the hardcore BH model with a superlattice [25], and its results showed an excellent agreement with Monte Carlo simulations [23, 25]. Motivated by the success of this tech- nique with these models, here we apply it to the two-species BH model, hoping to develop an analytical approach which could be as accurate as the numerical ones. It is the recent observation of this transition in effectively three- [2], one- [3], and two-dimensional [4, 5] optical lat- tices, which has been considered one of the most remarkable achievements in the field of ultracold atomic gases, since it paved the way for studying other strongly correlated phases in similar setups. Such lattices are created by the intersection of laser fields, and they are nondissipative periodic potent ial energy surfaces for the atoms. Motivated by this success in experimentally simulating the single-species BH model with ultracold atomic Bose gases loaded into optical lattices, there has been recently an intense theoretical activity in analyzing BH as well as Fermi-Hubbard type models [6]. The remaining paper is organized as follows. After in- troducing the model Hamiltonian in Sec. II, we develop the strong-coupling expansion in Sec. III, where we derive an analytical expression for the phase boundary between the in- compressible Mott insulator and the compressible superflui d phase. Then, in Sec. IV, we propose a chemical-potential ex- trapolation technique based on scaling theory to extrapolate our third-order power-series expansion into a functional form that is appropriate for the Mott lobes, and use it to obtain typ- ical ground-state phase diagrams. A brief summary of our conclusions is given in Sec. V. For instance, in addition to the Mott insulator and single- species superfluid phases, it has been predicted that the two - species BH model has at least two additional phases: an in- compressible super-counter flow and a compressible paired superfluid phase [7–16]. Our main interest here is in the latt er phase, where a direct transition from the Mott insulator to the paired superfluid phase (superfluidity of composite bosons, i.e. Bose-Bose pairs) has been predicted, when both species have integer fillings and the interspecies interaction is su ffi- ciently large and attractive. Given that the interspecies inter- actions can be fine tuned in ongoing experiments, e.g. with $^{41}$K-$^{87}$Rb with mixtures [17, 18], via using Feshbach reso- nances, we hope that some of our results could be tested with ultracold atomic systems. To describe Bose-Bose mixtures loaded into optical lattices, we consider the following two-species BH Hamiltonian, II. TWO-SPECIES BOSE-HUBBARD MODEL H = − ∑ i,j,σ t$_{ij,σ}$b † $_{i,σ}$b$_{j,σ}$ + ∑ i,σ U$_{σσ}$ 2 ̂ n$_{i,σ}$ $^{(}$̂ n$_{i,σ}$ − 1) + U$_{↑↓}$ ∑ i ̂ $^{n$_{i,}$$_{↑}$}$̂ n$_{i,}$$_{↓}$ − i,σ $^{µ$_{σ}$}$̂ n$_{i,σ}$, (1) arXiv:1001.0021v3[cond-mat.quant-gas]8 Oct 2010 I. INTRODUCTION