J.A. Plascak

Mean field solution of the general spin Blume-Capel model

We study the mean field solution of the general Blume-Capel model with integer and semi-integer spins. For integer spins there exist one tricritical point and a disordered phase at low temperatures which are not present for semi-integer spins. In both cases we find, at T=0, a multiphase point in which the different phases spread out when the temperature is increased. This fact gives origin to first-order transition lines between different ordered phases (characterized by different values of the magnetizations) each one terminating at an isolated critical point.

Multicritical points in the ferromagnetic binary Ising model

The phase diagram and the temperature dependence of the magnetization of the random-site binary ferromagnetic Ising model consisting of spin-12 and spin-1 components in the presence of crystal-field interactions is investigated by the use of a mean field theory. By considering in this simple system all the exchange interactions positive, seven topologically different types of phase diagrams are achieved, including a variety of multicritical points such as tricritical, fourth-order, critical end, and isolated critical. An ordered phase persisting for large values of the crystal-field interaction is also observed.

Ising model in a transverse random field

Abstract The transverse random-field Ising model with a trimodal distribution is studied within mean-field and mean-field renormalization-group approaches. The phase diagram is obtained and all the transition lines are second order. An ordered phase persists for large random fields provided that the probability of the zero transverse field is greater than the site-percolation threshold.

Phase transition in the three-dimensional anisotropic Heisenberg antiferromagnetic model☆

Abstract The anisotropic Heisenberg model with spin - 1 2 and antiferromagnetic (AF) interaction is studied by the mean field renormalization group (MFRG) approach on a simple cubic lattice. This model in the ferromagnetic (F) limit was treated previously in the literature by Plascak [J. Phys. A 17 (1984) L597], but in this work we generalize the MFRG method for the model with AF interaction. The critical temperature and critical exponent are obtained as a function of the anisotropy parameter (Δ). The results obtained are compared with the critical behavior of the F model. We observe that TN is higher than TC, but the F and AF case present the same universality class.

Renormalization group study of compressible Ising systems

Abstract The mean field renormalization group approach is used to study compressible Ising models. The phase diagram as well as estimates of critical exponents are obtained for systems where shear forces are neglected.

On the equivalence of finite size scaling renormalization group and phenomenological renormalization

It is shown that the recently proposed finite size scaling renormalization group, when using systems infinite in one dimension and finite in the others, is equivalent to the Nightingale (correlation length) phenomenological renormalization. The equivalence, however, is concerned only with the critical coupling and thermal critical exponent; the finite size scaling renormalization group approach provides other exponents in a more precise and elegant fashion through the flux diagram of the recursion relations in the space spanned by the parameters of the Hamiltonian. In addition, a new set of magnetic exponents, which are so accurate as the thermal ones, can now be obtained in an easier way.

Some exact results from the mean field renormalization group

Abstract The mean field renormalization group is applied to the study of the directed self-avoiding random walk in d dimensions and to the spin-½ ferromagnetic Ising model on the Bethe lattice. In both cases the known exact results are obtained.

Ferromagnetic critical line of the Ising model with crossing bonds

Abstract The ferromagnetic critical line of the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions is studied using the mean field renormalization group approach. The expected asymptotic behavior for large values of the nearest-neighbor interaction is achieved and the results are compared to those obtained from other methods.

Mean field renormalization group for the spin-12 random anisotropic Heisenberg model

Abstract The mean field renormalization group approach is applied to the spin- 1 2 random anisotropic Heisenberg mode. In this model the couplings between any pair of nearest-neighbour spins can be randomly isotropic (Heisenberg-like) or randomly anisotropic (Ising-like). The phase diagram and estimates of critical exponent are obtained for the two-and three-dimensional lattices.

A new mean-field-like renormalization group transformation

A new real space renormalization group transformation combining ideas from mean-field and finite-size scaling theories is presented. Application to the two-dimensional site directed percolation problem gives better values for the percolation threshold and critical exponent of longitudinal correlation length than those obtained previously with other real space renormalization group approaches. When applied to one-, two- and three-dimensional) Ising models, the results are comparable to the ones from previous mean-field-like renormalization group transformations. This method can easily be applied to other systems having second-order phase transitions.