The competition between cluster spin glass (CSG) and ferromagnetism or antiferromagnetism is studied in this work. The model considers clusters of spins with short-range ferromagnetic or antiferromagnetic (FE-AF) interactions (J_{0}) and long-range disordered couplings (J) between clusters. The problem is treated by adapting the correlated cluster mean-field theory of D. Yamamoto [Phys. Rev. B 79, 144427 (2009)]. Phase diagrams T/J×J_{0}/J are obtained for different cluster sizes n_{s}. The results show that the CSG phase is found below the freezing temperature T_{f} for lower intensities of J_{0}/J. The increase of short-range FE interaction can favor the CSG phase, while the AF one reduces the CSG region by decreasing the T_{f}. However, there are always critical values of J_{0} where AF or FE orders become stable. The results also indicate a strong influence of the cluster size in the competition of magnetic phases. For AF cluster, the increase of n_{s} diminishes T_{f} reducing the CSG phase region, which indicates that the cluster surface spins can play an important role in the CSG arising.

The phase diagrams of the three-state Ghatak-Sherrington spin-glass (or random Blume-Capel) model are obtained in mean-field theory with replica symmetry in order to study the effects of a ferromagnetic bias and finite random connectivity in which each spin is connected to a finite number of other spins. It is shown that inverse melting from a ferromagnetic to a low-temperature paramagnetic phase may appear for small but finite disorder and that inverse freezing appears for large disorder. There can also be a continuous inverse ferromagnetic to spin-glass transition. Inverse melting appears as the reversible first-order transition between a liquid or completely disordered paramagnetic phase at low temperature and a crystalline or ordered phase at higher temperature, whereas inverse freezing is the reversible first-order transition from a paramagnetic phase to a hightemperature amorphous or glassy phase [4,6]. These transitions usually appear with a reentrance in the phase boundary of continuous transitions at high temperature between an ordered and a fully disordered phase. The characteristic feature of inverse transitions is that the ordered high-temperature phase is favored by the entropy while the low-temperature disordered phase is favored by the minimum of the energy. This is best illustrated by the change of the folded into the unfolded configurations of methyl cellulose polymer chains in water. The bundles of methyl groups that are folded in a compact weakly interacting configuration at low T unfold with increasing T making more microscopic configurations available with an increase in volume and entropy [4,9].The three-state spin-glass (SG) model with a crystal-field term of Ghatak and Sherrington (GS) [1] is a Blume-Capel (BC) model [10,11] with random bonds that exhibits a continuous transition between a spin-glass and a paramagnetic (P) phase at high temperature and a reentrant phase boundary at low temperature. Inverse freezing appears on the latter as a * rubem@if.ufrgs.br † theumann@if.ufrgs.br ‡ sgmagal@gmail.com genuine thermodynamic first-order transition below a tricritical point in mean-field theory with infinite-range interactions and full replica symmetry breaking (FRSB) [3]. Similar results were obtained in mean-field theory with one-step replica symmetry breaking for the BC model with spin degeneracy [4].Inverse freezing also appears in numerical simulations for a three-dimensional GS model with nearest-neighbor interactions [6] and numerical work on the two-dimensional randombond Ising model exhibits an inverse melting transition [12]. It would be interesting to have independent analytical results for either of these random systems with finite-range interactions exhibiting both IT.Theoretical works on inverse freezing deal usually with a symmetric distribution of random bonds in fully connected systems. The purpose of the present work is to study, by means of an analytical procedure combined with a numerical evolution of a population dynamics [17], the dependence on disorder of the phase d...