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Physical Review Letters

Critical Phenomena in the Two-Dimensional XY Model Using a Real-Space Renormalization Group

Douglas M. Lublin

Published Mar 10, 1975• 31 citations
Publisher

Abstract

The phase transition of the classical $\mathrm{XY}$ model on a triangular lattice is studied with a renormalization-group technique based on the cumulant-expansion method of Niemeijer and van Leeuwen. By utilization of some approximations, a nontrivial fixed point is located, and this yields the following critical values: $\frac{J}{k{T}_{c}}=0.25$, $\ensuremath{\nu}=1.35$, and $\ensuremath{\eta}=0.21$.

DOI
10.1103/physrevlett.34.568

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The two-dimensional XY\mathrm{XY}XY and Heisenberg models are studied with a quantum-mechanical generalization of the Niemeijer and Van Leeuween renormalization-group method. In second-order cumulant expansion a fixed point exists for the XY\mathrm{XY}XY model, giving a critical temperature at $\frac{J}{k{T}_{c}}\ensuremath{\cong}0.87\ifmmode\pm\else\textpm\fi{}0.05$. For the Heisenberg model we find a fixed point at K∗=2.8{K}^{*}=2.8K∗=2.8. This result may not be reliable, since the truncation of the cumulant expansion is only valid for small KKK.
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-12\frac{1}{2}21​ $X\ensuremath{-}Y$ 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.
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