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 interacti
on, 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 $z+etasim1$. Unusually small triplet and singlet excitation energies found in extended points of the Brillouin zone impose constraints on this algebraic spin liquid.
Motivated by a recent experiment on volborthite, a typical spin-$1/2$ antiferromagnet with a kagom{e} lattice structure, we study the magnetization process of a classical Heisenberg model on a spatially distorted kagom{e} lattice using the Monte Carl
o (MC) method. We find a distortion-induced magnetization step at low temperatures and low magnetic fields. The magnitude of this step is given by $Delta m_z=left|1-alpharight|/3alpha$ at zero temperature, where $alpha$ denotes the spatial anisotropy in exchange constants. The magnetization step signals a first-order transition at low temperatures, between two phases distinguished by distinct and well-developed short-range spin correlations, one characterized by spin alignment of a local $120^{circ}$ structure with a $sqrt{3}timessqrt{3}$ period, and the other by a partially spin-flopped structure. We point out the relevance of our results to the unconventional steps observed in volborthite.
