  We study the magnetic properties of the $t-J$ model on triangular lattice in
light of the recently discovered superconductivity in Na$_{x}$CoO$_{2}$ system.
  We formulate the problem in the Schwinger Boson - slave Fermion scheme and
proposed a sound mean field ansatz(canting ansatz) for the RVB order
parameters. Working with the canting ansatz, we map out the temperature-doping
phase diagram of the model for both sign of the hopping term. We find the
prediction of the $t-J$ model differ drastically from that of earlier LSDA
calculation and there is large doping range in which the system show zero net
magnetization, rather than saturated magnetization as predicted in the LSDA
calculation. We show the result of LSDA is unreliable in the strong coupling
regime due to its neglect of electron correlation. We find the spin Berry phase
play a vital role in this geometrically frustrated system and the various
states in the phases diagram are characterized(and distinguished) by their
respective spin Berry phase, rather than any Landau-like order parameter
related to broken symmetry. We find the spin Berry phase is responsible for the
qualitative difference in the low energy excitation spectrum of the various
states of the phase diagram. We argue the phase boundary in the mean field
phase diagram may serve as the first explicit and realistic example for phase
transition between states with different quantum orders which in our case is
nothing but the spin Berry phase. We also find an exotic state with nonzero
spin chirality but no spin ordering is stable in a large temperature and doping
range and find.......
