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Journal of Physics C Solid State Physics

Critical curve of the highly anisotropic two-dimensional spin-s Ising model

Leigh Sneddon, Robin B. Stinchcombe

Published Oct 14, 1978• 4 citations
Publisher

Abstract

The rectangular spin-s Ising ferromagnet with nearest-neighbour interactions is considered. The interaction energies are of the form s -2 J sigma i sigma j and s -2 J' sigma k sigma l , respectively for bonds in the two lattice directions, where the spin variables sigma i can take the values -s, -s+1,...,s-1,s. A transfer matrix formalism is used. By relating the largest eigenvalues of the transfer matrix for spin-s to those for spin- 1 / 2 , the leading order form for J'<<J of the spin-s critical curve is investigated. Subject to plausible assumption concerning the thermodynamic limit, the critical curve is shown to be given by K'=2s exp(-2K)(1+ epsilon ) where K=J/kT, K'=J'/kT and epsilon to 0 as K to infinity .

DOI
10.1088/0022-3719/11/19/014

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The rectangular spin-s Ising ferromagnet with nearest-neighbour interactions is considered. The interaction energies are of the form s -2 J sigma i sigma j and s -2 J' sigma k sigma l , respectively for bonds in the two lattice directions, where the spin variables sigma i can take the values -s, -s+1,...,s-1,s. A transfer matrix formalism is used. By relating the largest eigenvalues of the transfer matrix for spin-s to those for spin- 1 / 2 , the leading order form for J'<<J of the spin-s critical curve is investigated. Subject to plausible assumption concerning the thermodynamic limit, the critical curve is shown to be given by K'=2s exp(-2K)(1+ epsilon ) where K=J/kT, K'=J'/kT and epsilon to 0 as K to infinity .
A phenomenological renormalisation transformation is used to calculate critical properties of two-dimensional spin s Ising systems with lattice anisotropy. Systems with s= 1 / 2 , 1, 3 / 2 , 2, 5 / 2 are considered for a large range of ratios, R=K y /K x , of exchange constants in the x and y spatial directions. Agreement with the exact solution for s= 1 / 2 is good, with the percentage error in T c becoming very small as R approaches zero. The results for nu are consistent with universality arguments. The estimates of the critical curves suggest an asymptotic form, as R to 0, of K y approximately 2s exp(-2K x ).
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r -wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K x and K y (where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K z, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of anisotropic two-layer Ising ferromagnet, on strips of r wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest neighbor couplings Kx, Ky (where Kx and Ky are the nearest neighbor interactions within each layer in the x and y directions, respectively) and with inter-layer coupling Kz, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted on a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained, are in good agreement with the accurate values reported by others.
The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.