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Green function metheod is introduced to investigate ultracold dilute gas of bosonic atoms in an optical lattice which can be described by a Bose–Hubbard model. The superfluid–Mott insulator phase transition condition is determined by the related energy-band structure with an obvious interpretation of the transition mechanism.

Within a mean-field treatment of the Bose–Hubbard model for an optical lattice, we have derived a self-consistent equation for the order parameter of possible phases in the optical lattice at finite temperatures. From the solutions to the self-consistent equation, we have inferred the temperature dependence of the order parameter and transition temperatures of Mott-insulator and superfluid phases into the normal phase. The condensation fraction in the superfluid phase has been deduced from the one-body density matrix and its temperature dependence has been given. In terms of the normalized correlation function of quasiparticles, strong coherence in the superfluid phase and its loss in Mott-insulator phases are demonstrated.

The advent of coherent matter waves in the form of Bose-Einstein condensates, coupled with periodic potentials in the form of optical lattices, has established a new area of research on the boundary between atomic and condensed matter physics. This article is a brief review of the recent experimental progress in the area of degenerate Bose gases loaded into optical lattices, including strongly correlated systems and the role of dimensionality. Future prospects will also be outlined.

In this paper, a new Bosonic Gutzwiller projection approach is proposed to study the strongly correlated bosons in optical lattice. In this method, there exist many variational parameters which make us calculate the physical characters of states, including the double occupation rate and the higher occupation rates. Based on this approach, a quantum phase transition from superfluid state to Mott insulator state is obtained for the homogenous phase at unit filling.

We use the Density Matrix Renormalization Group and the Abelian bosonization method to study the effect of density on quantum phases of one-dimensional extended Bose–Hubbard model. We predict the existence of supersolid phase and also other quantum phases for this system. We have analyzed the role of extended range interaction parameters on solitonic phase near half-filling. We discuss the effects of dimerization in nearest neighbor hopping and interaction as well as next nearest neighbor interaction on the plateau phase at half-filling.

The finite size scaling behavior of superfluid–insulator transition in the one-dimensional Bose–Hubbard model is studied. It is shown that the superfluid density of the system with finite size has a maximum at a certain interaction U_m and the derivative of superfluid density has a minimum at a certain interaction U_d. The critical point U_c can be quantified by the scaling analysis of either U_m or U_d. The transition point U_m tends to the critical point U_c from the region of U < U_c, while the U_d tends to the U_c from the region of U > U_c. The transition points U_m and U_d satisfy different finite size scaling laws and have the different critical exponents. The divergence speed of the superfluid density is much smaller than that of its derivative at the critical point.

A self-consistent equation is derived for the order parameter of quantum phases in an optical lattice within a mean-field approximation to the Bose–Hubbard model. Analyzing the solutions of the self-consistent equation in terms of the number fluctuations of quasiparticles, the one-body density matrix of atoms, and the normalized correlation function of quasiparticles, we have identified two types of Mott-insulator phases that are characterized by different values of the order parameter, quasiparticle fluctuations, and correlations. We have also identified the quantum critical points separating these two types of Mott-insulator phases and the corresponding quantum critical lines on the phase diagram.

In this paper, we study the ultracold atoms in optical lattice with a weak random external potential by an extended Bose–Hubbard model. When the on-site interaction is strong enough, the model can be mapped to the XXZ model. Then the mean-field theory is applied and we get the zero- and finite-temperature phase diagrams in different optical parameters. The differences between the systems with and without disorder were found, and the Bose-glass phase may exist in the system with disorder.

The spin dynamics of ultracold bosonic atoms in a double well at finite temperatures is studied in terms of the spin structure factor within the two-site Bose–Hubbard model. The spin structure factor for positive frequencies reveals spin excitations at two frequencies corresponding two transitions among the eigenstates of the Hamiltonian. It has been found that the temperature effect is quite substantial. At zero temperature, only spin excitations at the low frequency survive, while the spectral weights at the two frequencies become approximately equal at high temperatures. As temperature increases, the spectral weight shifts from the low frequency to the high frequency.

Topological phases are important topics in condensed matter physics. Here, we investigate a spin–orbit coupling Bose–Hubbard model in triangular lattice. In the strong coupling limit, we obtained the single particle Green’s function and constructed the phase diagram for ground states. We found two types of nontrivial topological ground phases characterized by spin Chern number and skyrmion number, respectively. The spin Chern numbers characterize the spin Chern insulators. While the skyrmion numbers characterize the skyrmion textures. We show that the phase transitions between different spin Chern insulators take place with gap closing and reopening. While the phase transitions between different skyrmion textures occur without gap closing.

The advent of coherent matter waves in the form of Bose-Einstein condensates, coupled with periodic potentials in the form of optical lattices, has established a new area of research on the boundary between atomic and condensed matter physics. This article is a brief review of the recent experimental progress in the area of degenerate Bose gases loaded into optical lattices, including strongly correlated systems and the role of dimensionality. Future prospects will also be outlined.