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We study the stability of fixed points in the two-loop renormalization groupfor the random field O($N$) spin model in $4+\epsilon$ dimensions. We solve thefixed-point equation in the 1/N expansion and $\epsilon$ expansion. In thelarge-N limit, we study the stability of all fixed points. We solve theeigenvalue equation for the infinitesimal deviation from the fixed points underphysical conditions on the random anisotropy function. We find that the fixedpoint corresponding to dimensional reduction is singly unstable and others areunstable or unphysical. Therefore, one has no choice other than dimensionalreduction in the large-N limit. The two-loop $\beta$ function enables us tofind a compact area in the $(d, N)$ plane where the dimensional reductionbreaks down. We calculate higher-order corrections in the 1/N and $\epsilon$expansions to the fixed point. Solving the corrected eigenvalue equationnonperturbatively, we find that this fixed point is singly unstable also forsufficiently large $N$ and the critical exponents show a dimensional reduction.Comment: 9 pages, 2 figure

physical review. b, condensed matter and materials physics

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Stability of fixed points in the(4+ϵ)-dimensional random fieldO(N)spin model for sufficiently large