Real-space renormalization group and critical phenomena in the two-dimensional spin-12X−Ymodel

A new formulation of the renormalization-group in real space, suitable for quantum spin systems is proposed. The method is applied to the two-dimensional spin-$\frac{1}{2}$ $X-Y$ model on a triangular lattice, the renormalization-group transformation being evaluated up to second order in an appropriate cumulant expansion. To first order an unstable fixed point of the transformation is found, corresponding to a critical temperature and critical indices in satisfactory qualitative agreement with present high-temperature series expansion estimates. In the second-order calculation, however, this fixed point disappears, thus throwing some doubt on the conventional picture of criticality as furnished by high-temperature series. The free energy of the model is also computed. For relatively small values of the nearest-neighbor coupling it is in good agreement with that found by high-temperature series analysis.