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Physical review. B, Solid state

Critical temperatures of the spin- s Ising model

Theodore W. Burkhardt, Robert H. Swendsen

Published Apr 1, 1976• 38 citations
Publisher

Abstract

Critical temperatures of the spin-$s$ 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-$s$ Hamiltonian into a spin-1/2 Hamiltonian, which is then analyzed with a spin-1/2 renormalization transformation. The calculation is performed for the $d=2$ square lattice and the $d=3$ 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.

DOI
10.1103/physrevb.13.3071

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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.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
The high-temperature susceptibility of the spin-SSS Ising model on the triangular net is analyzed as a function of SSS using tenth-order high-temperature series expansions. In particular the spin dependence of the critical temperature and the amplitude of the dominant singularity are obtained by assuming a universal value $\ensuremath{\gamma}=\frac{7}{4}$ for the critical exponent of the susceptibility. The results, for the critical temperature particularly, provide a valuable guidepost in judging the convergence of recent nonlinear renormalization-group calculations for these models.
Describes the development of high-temperature series expansions for general mixed spin-S-spin-S' Ising models on simple loose-packed lattices. It extends previous work on the simpler special case S'=1/2. Coefficients of the series for the initial susceptibility and specific heat of these models on the square, SC and BCC lattices are presented. A preliminary analysis of the results for the particular case S=1, S'=3/2 is given, and the critical ratio K c and the susceptibility exponent gamma are estimated.
The fixed-point topology of Kadanoff's lower-bound renormalization transformation is discussed for the d=2d=2d=2 square-lattice Ising model in the three-dimensional space of coupling constants for the nearest-neighbor, next-nearest neighbor, and four-spin interactions. An obstacle one encounters in trying to calculate a critical surface is pointed out. In the three-dimensional space of coupling constants Kadanoff's fixed point has two relevant vectors and can only be reached from a critical line, not from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. Another fixed point is reported which does have only one relevant vector and can be reached from a critical surface. The critical exponents of this fixed point are close to the exact values, though not nearly so close as for Kadanoff's fixed point. The properties of both fixed points as a function of the variational parameter ppp are described. Critical values of the nearest-neighbor couplings leading to the fixed points are compared.