  We numerically study the Heisenberg models on triangular lattices by
extending it from the simplest equilateral lattice with only the
nearest-neighbor exchange interaction. We show that, by including an additional
weak next-nearest-neighbor interaction, a quantum spin-liquid phase is
stabilized against the antiferromagnetic order. The spin gap (triplet
excitation gap) and spin correlation at long distances decay algebraically with
increasing system size at the critical point between the antiferromagnetic
phase and the spin-liquid phase. This algebraic behavior continues in the
spin-liquid phase as well, indicating the presence of an unconventional
critical (algebraic spin-liquid) phase characterized by the dynamical and
anomalous critical exponents $z+\eta\sim1$. Unusually small triplet and singlet
excitation energies found in extended points of the Brillouin zone impose
constraints on this algebraic spin liquid.
