The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.
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 Ising spin glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture that relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Toward this end we precisely determine the global phase diagrams for these six hierarchical spin glasses, using up to $2.5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities
d and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with
d=2 or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.
We study the criticality at the spin-glass-ferromagnetic-paramagnetic multicritical point in the d=3, \ifmmode\pm\else\textpm\fi{}J distribution, random-bond Ising model. Using high-temperature expansions to order ${\mathit{T}}^{\mathrm{\ensuremath{-}}34}$, we estimate that the multicritical point N lies on the Nishimori line at
Tc/J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.
We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [$tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1$ for the $\ifmmode\pm\else\textpm\fi{}J$ distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori's manifold is expected to control multicriticality for a wider class of models.