Random fields, random anisotropies, nonlinearσmodels, and dimensional reduction

The effects of weak random fields and random higher-rank anisotropies are investigated for n-component magnets with n≥2 for low temperatures near four dimensions. All of the random fields and anisotropies are marginal at the lower critical dimension, $d_c$=4. By use of supersymmetry, it is shown that at zero temperature the formal perturbation expansion in powers of the strengths, {${\Delta }_{\mu }$}, of the random fields and all the anisotropies only depend on a simple combination, Δ̃, of the {${\Delta }_{\mu }$}. Furthermore, this expansion in powers of Δ̃ is equivalent to the expansion of the pure n-component system in powers of T in two dimensions less. It would be natural to conclude that this implies a generalized dimensional reduction for exponents; however, we present renormalization-group calculations which indicate that this is not valid. Functional renormalization-group recursion relations are derived which couple together all the random fields and anisotropies. It is demonstrated that, in contrast to earlier claims, there is no perturbative fixed point of the renormalization group in 4+ε dimensions. The flows go into regimes where nonperturbative effects are important and it is argued that dimensional reduction is likely to break down. This is the first example of which we are aware of a problem for which an infinite number of marginal operators play an important role. Various suggestions concerning the behavior of random-anisotropy magnets are discussed in the light of the present results.