The mixture of the scalar bosonic and the spinless or polarized fermionic cold atoms in the one-dimensional optical lattice is studied. The system is modeled by the Bose-Fermi-Hubbard Hamiltonian, which shows different behavior from that of the Bose-Hubbard or the Fermi-Hubbard models. Because the $\text{SU}(1\ensuremath{\mid}1)$-supersymmetric Bethe ansatz solution gives an excellent approximation to this kind of mixed cold atomic systems, the ground-state properties of the system such as the densities of state in the momentum space are obtained based on the Bethe ansatz. If the number of bosons is equal to that of fermions and the filling factor is 1, it is found that there exists a critical on-site interaction
Uc. If
U<Uc, the ground state of the system is in the superfluid phase, while if
U>Uc, the ground state is in the insulating phase. The superfluid-insulator transition occurs at
Uc. From the analysis of the superfluid density, the value of the critical point is determined as
Uc=2.79256, which is larger than the
Uc=0 for the Fermi-Hubbard model and smaller than the
Uc=3.28 for the Bose-Hubbard model. The elementary excitations and effective Hamiltonian in the strong-coupling limit are also discussed.
The quantum phase transition in binary Bose Gases is studied using the Cornwall-Jackiw-Tomboulis effective potential approach in the double-bubble approximation, which preserves the Goldstone theorem. Its main feature is that the transition is second order occurring at ultra-cold temperatures associated with the type of inverse symmetry-breaking transition occurring when the chemical potential reaches a critical value. However, it cannot simultaneously occur for the two components of a binary mixture of Bose gases.
Ultracold atomic gases may be the ultimate quantum simulator.These isolated systems have the lowest temperatures in the observable universe, and their properties and interactions can be precisely and accurately tuned across a full spectrum of behaviors, from few-body physics to highly-correlated many-body effects.The ability to impose potentials on and tune interactions within ultracold gases to mimic complex systems mean they could become a theorist's playground.One of their great strengths, however, is also one of the largest obstacles to this dream: isolation.This thesis touches on both of these themes.First, methods to characterize phases and quantum critical points, and to construct finite temperature phase diagrams using experimentally accessible observables in the Bose Hubbard model are discussed.Then, the transition from a weakly to a strongly interacting Bose-Fermi mixture in the continuum is analyzed using zero temperature numerical techniques.Real materials can be emulated by ultracold atomic gases loaded into optical lattice potentials.We discuss the characteristics of a single boson species trapped in an optical lattice (described by the Bose Hubbard model) and the hallmarks of the quantum critical region that separates the superfluid and the Mott insulator ground states.We propose a method to map the quantum critical region using the single, experimentally accessible, local quantity R, the ratio of compressibility to local number fluctuations.The procedure to map a phase diagram with R is easily generalized to inhomogeneous systems and generic many-body Hamiltonians.We illustrate it here using quantum Monte Carlo simulations of the 2D Bose Hubbard model.Secondly, we investigate the transition from a degenerate Fermi gas weakly coupled List of Tables Table Page 2.1 Techniques to Probe Atoms in Optical Lattices . . . . . .