Studies of magnetism and superconductivity in the triangular lattice

This thesis project is mainly inspired by the newly discovered superconductor NaxCoO2 · yH 2O. It has a layered structure with an almost separated single band out of the inner band electrons. It becomes the theoretical focus because it is the first layered strongly correlated system that shows superconductivity besides the cuprate superconductors. It may help people to understand the superconductivity in the cuprate system. In this thesis we mainly focus our attention on the strongly correlated models on triangular lattice. These models are expected to provide a theoretical background for the Nax CoO2 · yH2O system. We investigate the magnetic and superconductivity properties of these models on the triangular lattice. By the mean field calculation of different magnetic orders we got a phase diagram of the triangular lattice Hubbard model for different magnetic orders in the Hubbard model in the triangular lattice. To further investigate the superconductivity pairing symmetry of the triangular lattice, we use the standard variational Monte Carlo method to find which kind of pairing symmetry is stabilized in the triangular lattice T -- J -- V model. Our finding is that the extended- f wave pairing is most stable in the lattice.;To further understand the properties of the related model in the electron interacting system in the triangular lattice, we also solve the few particle problem in the triangular lattice. For two particles with the T -- U -- J -- V model in the triangular lattice, we got the exact solutions. For four particles of the Hubbard model in the triangular lattice we got the asymptotic solutions. We discuss the existing of the bound states in these solutions and the pairing symmetry of these bound states. For system with two-hole and one down-spin away from the all up spin background we also get a solution utilizing a large U expansion. We discussed the relation of this solution with the Nagaoka states.;The third part of this thesis is on one hole problem of asymmetric T and T -- Jz model, which we found can be turned into a single body problem and is solvable. We get the spectrum of the model and calculate as a result the mass of the hole. This model may be related to the two species fermion coexisting system which may be realized in the optical lattice.