The Quantum Phase Transition Of Ultraclod Bose-Femi Mixture Gas In Optical Lattice

Research of the ultra-cold atom gas can help us to understand the nature of quantum many-body system. The realization of degenerate fermion gas and the BEC of dilute alkalis metal atoms open up a new direction for the study of ultra-cold atom. Making use of the technology of laser cooling and magnetic field capture, we can confine the ultra-cold atoms in the optical lattice, it have an advantage that adjust the interaction between two atoms and hopping, in order to control system's quantum phase translation. Besides, the system can be described by a very simple model and study the essential properties.As we know, the electronic interaction is compactness with the magnetism in solid matters, also the metal-insulation translation. In this paper, we use fermi-hubbard model describe the fermion system in one dimensional optical lattics,and simplify the model at some limit conditions, by study the relation of particles density and chemical potential, we found a jump of chemical potential when the particles density keep constant, this indicate the Mott insulation state is uncompressible. The system is stable when half filling, so it prefer to have single particle per site. As the interaction between particles increasing or the temperature decreasing, the local moment formate, so the system emerge magnetism.It is similar to the fermion, boson in optical lattices can described by bose-hubbard model, the competition between interaction energy and hopping energy of boson lead superfluid-Mott insulation phase translation. We use the strong-coupling method to study the model and calculate the energy of system, the result indicate that the Mott insulate state has energy gap between particle and hole excitation, when the system enter into superfluid state, the energy gap disappear, making the particles can hopping freely, in other words, the system have particle-hole symmetry. In addition, the distinction of the two state is particle density, if the particle number of single site is integer and invariable, the system is insulate, or else is superfluid.With the analytical method, we can get some qualitative results of the quantum phase transition, witch reliability can be judge by numerical calculation. In this paper, we use full diagonalization, Lanczos exact diagonalization and density matrix renormalization group to deal with the one dimensional lattice. Although the system include several sites, the result is agreed with the analyzing method. By calculating the thermodynamic quantity, we get the conclusion that the hopping of particles is the main reason witch lead to phase transition with low temperature, when the interaction of particles keep constant.