  The $120^0$ ordered antiferromagnetic state of the Hubbard model on a
triangular lattice presents an interesting case of $U$-controlled competing
interactions and frustration. The spin stiffness is found to vanish at $U^*
_{\rm stiff} \approx 6$ and the spin-wave energy $\omega_M$ at ${\bf
q}_M=(2\pi/3,0)$ etc. is found to vanish at $U_M ^* \approx 6.8$ due to
competing spin couplings generated at finite $U$. The loss of magnetic order
due to the magnetic instability at ${\bf q}_M$ yields a first-order quantum
phase transition in the insulating state at $U=U_{\rm M}^*$. Implications of
the quantum spin disordered insulator to the spin-liquid state and Mott
transition in the organic systems $\rm \kappa -(BEDT-TTF)_2 X$ are discussed.
Effects of hole and electron doping on magnetic ordering and spin stiffness are
also examined.
