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Journal of Physics C Solid State Physics

Critical properties of two-dimensional systems with randomly mixed Ising and Heisenberg bonds

Andrzej Pękalski

Published Dec 14, 1977• 6 citations
Publisher

Abstract

Using the high-temperature series expansion method the critical temperature and critical index gamma for a system consisting of a random mixture of Ising and Heisenberg couplings have been estimated. It has been found that gamma as a function of concentration p of Heisenberg-type bonds has a minimum the location of which depends strongly on the ratio of the Ising to Heisenberg coupling constants. It is conjectured that this might correspond to a change in the character of the transition.

DOI
10.1088/0022-3719/10/23/012

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Using the high-temperature series expansion method the critical temperature and critical index gamma for a system consisting of a random mixture of Ising and Heisenberg couplings have been estimated. It has been found that gamma as a function of concentration p of Heisenberg-type bonds has a minimum the location of which depends strongly on the ratio of the Ising to Heisenberg coupling constants. It is conjectured that this might correspond to a change in the character of the transition.
Using the high-temperature series expansion method, a system of randomly mixed ferromagnetic and antiferromagnetic Ising bonds is considered in quenched and annealed systems for an arbitrary value of spin and a BCC lattice. The dependence of the critical temperature on the concentration p of ferromagnetic bonds is determined. It is found that the resulting phase diagrams depend on the system (whether quenched or annealed), the spin value and the strength of the coupling constants. Also the dependence of the critical exponent gamma on p and the crossover exponent phi are estimated.
The two-dimensional Ising model with two kinds of bonds –ferro and antiferromagnetic– is investigated in both quenched and annealed approximations. High temperature series expansion technique is used to get the respective free energies and susceptibilities. Form the analysis of the ratio method results, the phase diagrams are drawn and critical concentrations are estimated. It is shown that in both approximations the system behaves similarly, although in the annealed case the magnetically ordered phases are more stable against increasing impurity concentration. It is also proved that for Bethe lattices initial susceptibilities for quenced and annealed systems coincide.
The method of high-temperature series expansions is used to study the dependence of T c on the concentration p of interactions in a randomly diluted S= 1 / 2 Heisenberg model. It is found that, for all three cubic lattices, the interval of p over which the Pade approximants to the series converge extends further towards the critical probability for percolation than for the corresponding site diluted model. In this interval the critical curve is linear and extrapolates to intersect the T=0 boundary at a value of p marginally above the critical probability for bond percolation. The anomalous dependence of the susceptibility exponent gamma on p which has recently been reported for the site diluted model is not observed for the model studied here.
The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.