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physica status solidi (b)

Effective Hamiltonian for the Two‐Dimensional Arbitrary Spin Ising Model

J. Szṅajd

Published Jul 1, 1984• 2 citations
Publisher

Abstract

Abstract A method of the reduction of the generalized arbitrary spin two‐dimensional Ising model to the spin‐half Ising model is presented. The method is demonstrated in detail by calculating the effective interaction constants to the third order in a cumulant expansion for the triangular spin‐one Ising model (the Blume‐Emery‐Griffiths model).

DOI
10.1002/pssb.2221240111

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Physical review. B, Condensed matter•May 1, 1985•36 citations
Abstract A method of the reduction of the generalized arbitrary spin two‐dimensional Ising model to the spin‐half Ising model is presented. The method is demonstrated in detail by calculating the effective interaction constants to the third order in a cumulant expansion for the triangular spin‐one Ising model (the Blume‐Emery‐Griffiths model).
For pt.I see ibid., vol.14, p.57 (1981). The recently developed theory of correlation reduction method near transition temperature is generalised to the Ising model with arbitrary spin. The generalisation from the case of spin 1 / 2 is natural and straightforward, although some non-trivial technicalities are involved for the detailed applications. The transition temperatures for the spin 1 and spin 3 / 2 Ising ferromagnets are obtained explicitly, and are shown to be consistent with the general trend exhibited by other calculations.
We have calculated the three-spin cell cumulant expansion of Niemeijer and Van Leeuwen for the one-dimensional XY\mathrm{XY}XY model in order to obtain the critical exponents of the two-dimensional Ising model. We found that the calculation of this type is not as accurate as the direct cumulant-expansion calculation for the two-dimensional Ising model. Specifically, the results of our present calculation deviate considerably from the exact results in comparison to the results obtained by Hsu, Niemeijer, and Gunton. Moreover, we find that the odd-spin eigenvalue is not physical in this particular renormalization-group transformation.
We have calculated the exact ground-state phase diagrams of the spin-3/2 Ising model using the method that was proposed and applied to the spin-1 Ising model by Dublenych (2005 Phys. Rev. B 71 012411). The calculated, exact ground-state phase diagrams on the diatomic and triangular lattices with the nearest-neighbor (NN) interaction have been presented in this paper. We have obtained seven and 15 topologically different ground-state phase diagrams for J >0 and J <0, respectively, on the diatomic lattice and have found the conditions for the existence of uniform and intermediate or non-uniform phases. We have also constructed the exact ground-state phase diagrams of the model on the triangular lattice and found 20 and 59 fundamental phase diagrams for J >0 and J <0, respectively, the conditions for the existence of uniform and intermediate phases have also been found.