We study the diluted site-bond-correlated (SBC) Ising model within the scheme of the mean-field renormalization group. The SBC Ising model has been proposed to describe the phase diagram for
KNip${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$
F3. One treats the square (Z=4) and simple cubic (Z=6) lattices by using two different exchange interactions (denoted by models A and B). In the present paper, the phase diagram for some values of the correlation parameter \ensuremath{\alpha}, which range from zero to unity, is studied. We obtain two critical concentrations
pc(\ensuremath{\alpha}=0) and
pc(\ensuremath{\alpha}\ensuremath{\ne}0) in agreement with results from Monte Carlo methods. In addition, an interesting universal relation between the initial slope ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=1) and ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=0) is obtained (\ensuremath{\mu} denotes model A or B). From this relation, we can estimate the possible exact values to the initial slope on the completely correlated case. Our approach is valid for lattices with any coordination number Z.
We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration
p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension
L up to
700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
The three-dimensional site-random Ising model consisting of two types of magnetic ions, A and B, is investigated by means of the exchange Monte Carlo method. The nearest-neighbor interactions are antiferromagnetic (AF) for the B-B pair and ferromagnetic (FM) otherwise. A uniform FM or AF ordering is found to exist in a wide range of concentration c of A ions, in contrast with the bond-random Ising model such as the ± J model. The phase diagram is found to be symmetric on the c – T plane, though the ratio of the FM bonds to AF ones is asymmetric with respect to c =1/2. Spin configurations of degenerate ground states and their cluster structure are also studied. Even in the system with c =0.55, there exists only a unique rigid cluster which includes about 90% of spins and that cluster shows nonvanishing uniform magnetization. These results strongly suggest that the present model has neither a spin glass phase nor an FM-AF mixed phase, though their existence in a quite narrow region around c =1/2 has not been completely excluded yet.
The phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic (
F), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the
F and the AF phases approach each other, and one may find regions in parameter space where a mixed
F-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic (
P),
F (or AF) and SG phases coexist (
P-SG-
F or
P-SG-AF), or where the
P,
F, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist (
P-SG-
F-AF). Recent renormalization-group analysis of
P-SG-
F, by Chen and Lubensky, yields complex exponents for the
XY and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for
P-SG-
F-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.
The anisotropic two-dimensional Ising model in the presence of a magnetic field is studied within two different approaches: the effective-field theory (EFT) with correlation and the Bethe-Peierls (BP) approximation. The model consists of ferromagnetic interaction
(Jx) in the
x direction and antiferromagnetic interaction
(Jy) in the
y direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case
Jx=Jy. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of
H∕J∊[0,2], with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of
r=Jy∕Jx, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.