A BRIEF ACCOUNT OF THE ISING AND ISING-LIKE MODELS: MEAN-FIELD, EFFECTIVE-FIELD AND EXACT RESULTS

摘要

The present article provides a tutorial review on how to treat the Ising and Ising-like models within the mean-field, effective-field and exact methods. The mean-field approach is illustrated on four particular examples of the lattice-statistical models: the spin-1/2 Ising model in a longitudinal field, the spin-1 Blume-Capel model in a longitudinal field, the mixed-spin Ising model in a longitudinal field and the spin-S Ising model in a transverse field. The mean-field solutions of the spin-1 Blume-Capel model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A continuous quantum phase transition of the spin-S Ising model driven by a transverse magnetic field is also explored within the mean-field method. The effective-field theory is elaborated within a single-and two-spin cluster approach in order to demonstrate an efficiency of this approximate method, which affords superior approximate results with respect to the mean-field results. The long-standing problem of this method concerned with a self-consistent determination of the free energy is also addressed in detail. More specifically, the effective-field theory is adapted for the spin-1/2 Ising model in a longitudinal field, the spin-S Blume-Capel model in a longitudinal field and the spin-1/2 Ising model in a transverse field. The particular attention is paid to a comprehensive analysis of tricritical point, continuous and discontinuous phase transitions of the spin-S Blume-Capel model. Exact results for the spin-1/2 Ising chain, spin-1 Blume-Capel chain and mixed-spin Ising chain in a longitudinal field are obtained using the transfer-matrix method, the crucial steps of which are also reviewed when deriving the exact solution of the spin-1/2 Ising model on a square lattice. The critical points of the spin-1/2 Ising model on several planar (square, honeycomb, triangular, kagome, decorated honeycomb, etc.) lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. The mapping transformation technique is subsequently adapted to obtain exact results for the critical temperature and spontaneous magnetization of the mixed-spin Ising model on decorated planar lattices and three-coordinated archimedean (honeycomb, bathroom-tile and square-hexagon-dodecagon) lattices. It is shown that an increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant phase transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than the critical temperature of the same model on two semi-regular archimedean lattices with the same coordination number. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is also examined in detail. Last but not least, the review affords a brief account of the Ising-like models previously solved within the mean-field, effective-field and exact methods along with a few comments on their future applicability.