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Physical review. B, Solid state

Critical interactions for the triangular spin- s Ising model by a spin-restructuring transformation

A. Nihat Berker

Published Oct 1, 1975• 29 citations
Publisher

Abstract

A transformation is constructed which maps the spin-$s$ Ising model onto a spin-\textonehalf{} Ising model with furtherneighbor and many-site interactions. Renormalization-group methods may thus be applied directly to the spin-$s$ problem by prefacing the usual sequence of position-space rescaling transformations with this spin-restructuring transformation. We demonstrate the method by calculating the critical interactions (i.e., the inverse critical temperatures) of the spin-$s$ Ising model on the triangular lattice, using the recent results of Niemeijer and van Leeuwen for the critical subspace of the corresponding spin-\textonehalf{} Ising model. Our evaluations agree very well with recent series data.

DOI
10.1103/physrevb.12.2752

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A transformation is constructed which maps the spin-sss Ising model onto a spin-\textonehalf{} Ising model with furtherneighbor and many-site interactions. Renormalization-group methods may thus be applied directly to the spin-sss problem by prefacing the usual sequence of position-space rescaling transformations with this spin-restructuring transformation. We demonstrate the method by calculating the critical interactions (i.e., the inverse critical temperatures) of the spin-sss Ising model on the triangular lattice, using the recent results of Niemeijer and van Leeuwen for the critical subspace of the corresponding spin-\textonehalf{} Ising model. Our evaluations agree very well with recent series data.
In evaluating the critical properties of lattice spin systems in the real-space renormalization-group theory we use the cluster variation method. A configuration in the transformed system is constrained and the probability of occurrence of this configuration is calculated both in the transformed system and in the original system. By equating the two probabilities and forming ratios of two such equalities (for two or more constrained configurations) the fixed point of the renormalization transformation is evaluated. The method can avoid the trouble due to different singularities in the original and transformed systems, and hence can obviate the possible development of spurious singularities in the transformation at low temperatures. The two-dimensional triangular Ising model is treated with numerical results comparable with those obtained by the cluster treatment of Niemeijer and van Leeuwen who used more and larger cluster types than those we introduce.
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.
Critical temperatures of the spin-sss Ising model with nearest-neighbor interactions are calculated with an approach based on Kadanoff's lower-bound renormalization transformation. An initial transformation converts the spin-sss Hamiltonian into a spin-1/2 Hamiltonian, which is then analyzed with a spin-1/2 renormalization transformation. The calculation is performed for the d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice. For comparison the critical temperatures are also estimated from the ten-term high-temperature series for the susceptibility given by Camp and Van Dyke. The agreement is very good.
Using a quantum version of the Niemeijer-van Leeuwen scheme we perform a real-space renormalization-group analysis for the Ising and XY models on a triangular lattice with inhomogeneous interactions. Due to the presence of different interactions we are able to consider as well the Kagomé lattice as a decorated triangular lattice in the limited cases. It is found that in this way we can investigate the flow diagrams and some thermodynamic properties of the models. For the Ising model the peaks in the specific heat are always connected with the presence of critical lines and are observed at the critical temperatures, while for the XY model their existence is not yet settled. For both models our results confirm universality with respect to inhomogeneous interactions.
The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.