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Physical review. B, Solid state

Renormalization-group approach to two-dimensional quantum models

Jos Rogiers, Raf Dekeyser

Published Jun 1, 1976• 46 citations
Publisher

Abstract

The two-dimensional $\mathrm{XY}$ 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 $\mathrm{XY}$ 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 ${K}^{*}=2.8$. This result may not be reliable, since the truncation of the cumulant expansion is only valid for small $K$.

DOI
10.1103/physrevb.13.4886

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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 JkTc\ensuremath≅0.87\ifmmode±\else\textpm\fi0.05\frac{J}{k{T}_{c}}\ensuremath{\cong}0.87\ifmmode\pm\else\textpm\fi{}0.05kTc​J​\ensuremath≅0.87\ifmmode±\else\textpm\fi0.05. 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.
The phase transition of the classical XY\mathrm{XY}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: JkTc=0.25\frac{J}{k{T}_{c}}=0.25kTc​J​=0.25, $\ensuremath{\nu}=1.35$, and $\ensuremath{\eta}=0.21$.
A quantum version of the Niemeijer--van Leeuwen real-space renormalization-group method is used to study the temperature dependence of the two- and four-spin correlations in the quantum XY model on the triangular lattice. The first-order cumulant expansion results suggest, similarly to other methods, a low-temperature phase of an essentially different kind from that predicted for the classical model. The possible explanation of the origin of the spurious 2D Heisenberg-like nontrivial fixed point in some renormalization-group calculations is also proposed.
The two-dimensional spin one half XY model may undergo an unusual type of phase transition. A formulation of the renormalization group approach of Niemeijer and van Leeuwen suitable for quantum-mechanical systems is presented. The method is applied to the spin one half XY model on the triangular lattice using a second order cumulant expansion based on division of the triangular lattice into seven spin cells. The resulting free energy curve is in excellent agreement with that obtained from the high temperature series expansion. An unstable fixed point is found but the corresponding critical temperature is not in good agreement with the series expansion value. Pathological values are found for the critical exponents.
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.
A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The $\ensuremath{\epsilon}$ expansion for nnn-component continuous spins shows that this behavior holds to first order in $\ensuremath{\epsilon}$ for n>4n>4n>4. For n<4n<4n<4, there is a new stable fixed point with $2\ensuremath{\nu}=1+[\frac{3n}{16(n\ensuremath{-}1)}]\ensuremath{\epsilon}$.