Validity of dimensional reduction in the random field O ( N ) spin model for sufficiently large N

We study the critical phenomena of a random field O(N) spin model near the lower critical dimension, by means of the renormalization group method and the 1/N expansion. We treat the O(N) nonlinear σ model including a random field and all the random anisotropy terms, and calculate the one-loop beta function for a linear combination of them in d = 4+ 2 under the assumption of replica symmetry. At first, we obtain all fixed points for the oneloop beta function in the large N limit, and discuss their stability. We find that the fixed point yielding dimensional reduction is singly unstable, and others are the fixed points with many relevant modes, or unphysical fixed point. Therefore, in the large N limit, the critical phenomena in 4 + 2 dimensions is found to be governed by the fixed point which gives the result of dimensional reduction. Next, we investigate the 1/N correction to the fixed point yielding dimensional reduction. Careful analysis of the eigenvalue equation for the infinitesimal deviation from the fixed point is done at order 1/N . In practice, the fixed point yielding dimensional reduction is found to be singly unstable. Thus, we conclude that the dimensional reduction holds for sufficiently large N .