A random-field Ising model that is capable of exhibiting a rich variety of multicritical phenomena, as well as a smearing of such behavior, is investigated. The model consists of an infinite-range-interaction Ising ferromagnet in the presence of a triple Gaussian random magnetic field, which is defined as a superposition of three Gaussian distributions with the same width σ, centered at H = 0 and H = ± H(0), with probabilities p and (1-p)/2, respectively. Such a distribution is very general and recovers, as limiting cases, the trimodal, bimodal and Gaussian probability distributions. In particular, the special case of the random-field Ising model in the presence of a trimodal probability distribution (limit [Formula: see text]) is able to present a rather nontrivial multicritical behavior. It is argued that the triple Gaussian probability distribution is appropriate for a physical description of some diluted antiferromagnets in the presence of a uniform external field, for which the corresponding physical realization consists of an Ising ferromagnet under random fields whose distribution appears to be well represented in terms of a superposition of two parts, namely a trimodal and a continuous contribution. The model is investigated by means of the replica method, and phase diagrams are obtained within the replica-symmetric solution, which is known to be stable for the present system. A rich variety of phase diagrams is presented, with one or two distinct ferromagnetic phases, continuous and first-order transition lines, tricritical, fourth-order, critical end points and many other interesting multicritical phenomena. Additionally, the present model carries the possibility of destroying such multicritical phenomena due to an increase in the randomness, i.e. increasing σ, which represents a very common feature in real systems.

Nearest-neighbor-interaction Ising spin glasses are studied on three different hierarchical lattices, all of them belonging to the Wheatstone-Bridge family. It is shown that the spin-glass lower critical dimension in these lattices should be greater than 2.32 . Finite-temperature spin-glass phases are found for a lattice of fractal dimension D ≈ 3.58 (whose unit cell is obtained from a simple construction of a part of the cubic lattice), as well as for a lattice of fractal dimension close to five. In the former case, the estimates of spin-glass critical temperatures associated with symmetric Gaussian and bimodal distributions are very close to recent results from extensive numerical simulations carried on a cubic lattice, suggesting that whole phase diagrams presented, obtained for couplings following non-centered distributions -not known up to the moment for Bravais lattices -should represent good approximations.

An Ising model with ferromagnetic nearest-neighbor interactions J 1 (J 1 > 0) and random next-nearest-neighbor interactions [+J 2 with probability p and −J 2 with probability (1 − p); J 2 > 0] is studied within the framework of an effective-field theory based on the differentialoperator technique. The order parameters are calculated, considering finite clusters with n = 1, 2, and 4 spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus p, for the particular case J 1 = J 2 , showing both superantiferromagnetic (low p) and ferromagnetic (higher values of p) orderings at low temperatures.Many magnetic compounds are well-described in terms of theoretical models characterized by a competition of nearest-neighbor and next-nearest-neighbor interactions. As typical examples, one has Eu x Sr 1−x S [1, 2] and Fe x Zn 1−x F 2 [3], which may present various low-temperature magnetic orderings, depending on its parameters, like the strength of these interactions and the concentration of magnetic ions x. The simplest model for such systems consists of an Ising model with competing uniform interactions, ferromagnetic J 1 (or antiferromagnetic −J 1 ) (J 1 > 0) on nearest-neighbor, and antiferromagnetic −J 2 (J 2 > 0) on next-nearest-neighbor spins (to be denoted hereafter as J 1 − J 2 Ising model).The J 1 −J 2 Ising model on a square lattice has attracted the attention of many workers, being investigated through several approximation methods ]. The various possible phases are the paramagnetic (P), for high temperatures, whereas for low temperatures one may have the ferromagnetic (F), antiferromagnetic (AF) and superantiferromagnetic (SAF) phases; the later is characterized by alternate ferromagnetic rows (or columns) of oppositely oriented spins. In the absence of a magnetic field, it can be shown that one has a symmetry with respect to the sign of the nearestneighbor interactions, i.e., the ferromagnetic and antiferromagnetic states are equivalent [5].At zero temperature (ground state), one has two ordered states depending on the value of the frustration parameter α = (J 2 /J 1 ), namely, the F (0 < α < 1/2) (in an equivalent way, an AF state occurs instead of the F one, if the sign of nearest-neighbor interactions is changed) and the SAF (α > 1/2) phases. On the other hand, its phase diagram for finite temperatures has been the object of some controversies. A continuous critical frontier between the P and F phases, for 0 < α < 1/2, with the critical temperature T c (α) decreasing and approaching zero for α → 1/2, is well-accepted nowadays. However, some characteristics of this phase diagram for α > 1/2 are polemic; in particular, the presence of a first-order transition between the P and SAF phases, as well as a tricritical point characterized by the coordinates (α t , T t ), beyond which a continuous critical frontier occurs, has been the object of some debate. Most of the works indicate that within the range 1/2 < α < α t one should get a first-order pha...

A nearest-neighbor-interaction Ising spin glass, in the presence of an external magnetic field, is studied on different hierarchical lattices that approach the cubic lattice.The magnetic field is considered as uniform, or random (following either a bimodal or a Gaussian probability distribution). In all cases, a spin-glass attractor is found, in the plane magnetic field versus temperature, associated with a low-temperature phase. The physical consequences of this attractor are discussed, in view of the present scenario of the spin-glass problem.

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