We study the critical behavior of a random field
O(N) spin model with a second-rank random anisotropy term in spatial dimensions
4<d<6, by means of the replica method and the
1∕N expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the
1∕N expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large
N limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large
N. Thus, we conclude that the dimensional reduction holds for sufficiently large
N.
The influence of a local anisotropy of random orientation on a ferromagnetic phase transition is studied for two cases of anisotropy axis distribution. To this end a model of a random anisotropy magnet is analyzed by means of the field theoretical renormalization group approach in two loop approximation refined by a resummation of the asymptotic series. The one-loop result of Aharony indicating the absence of a second-order phase transition for an isotropic distribution of random anisotropy axis at space dimension
d<4 is corroborated. For a cubic distribution the accessible stable fixed point leads to disordered Ising-like critical exponents.
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.
We reconsider the problem of the critical behavior of a three-dimensional
O(m) symmetric magnetic system in the presence of random-anisotropy disorder with a generic trimodal random-axis distribution. By introducing
n replicas to average over disorder it can be coarse grained to a ${\ensuremath{\phi}}^{4}$ theory with an $m\ifmmode\times\else\texttimes\fi{}n$ component order parameter and five coupling constants taken in the limit of $n\ensuremath{\rightarrow}0$. Using a field theory approach we renormalize the model to two-loop order and calculate the $\ensuremath{\beta}$ functions within the $\ensuremath{\varepsilon}$ expansion and directly in three dimensions. We analyze the corresponding renormalization group flows with the help of the Pad\'e-Borel resummation technique. We show that there is no stable fixed point accessible from physical initial conditions whose existence was argued in previous studies. This may indicate the absence of a long-range ordered phase in the presence of random-anisotropy disorder with a generic random-axis distribution.
The effects of Gaussian random fields applied only to the m spin fields of the anisotropic (n+m)-vector model are studied in the quenched case via the renormalized φ4 field theory by examining the stability of four fixed points and one fixed line in the large n and m limits. Regardless of the value of n/m, only a decoupled case is infrared stable, so that random fields prove to have no effects on the critical properties (e.g., critical indices) of the pure n spin fields. In such a decoupled case, as far as random fields are weak, longitudinal order corresponding to a random fixed point is observed remarkably. However, as the strength of random fields grows stronger, random fixed point moves to a Gaussian (trivial) fixed point and in consequence only transverse order gets conspicuous. The above results are physically consistent with Aharony's arguments.