Abstract
We study the stability of fixed points in the two-loop renormalization group
for the random field O($N$) spin model in $4+\epsilon$ dimensions.
We solve the fixed point equation in the $1/N$-expansion and $\epsilon$-expansion.
In the large-$N$ limit, we study the stability of all fixed points.
We solve the eigenvalue equation for the infinitesimal deviation from the fixed points
under physical conditions on the random anisotropy function.
We find that the fixed point corresponding to the dimensional reduction
is singly unstable and others are unstable or unphysical.
Therefore, one has no choice other than dimensional reduction in the large-$N$ limit.
The two-loop beta function enables us to find a compact area in $(d, N)$-plane
where the dimensional reduction breaks down.
We calculate higher order corrections in the $1/N$- and $\epsilon$-
expansions to the fixed point.
Solving the corrected eigenvalue equation nonperturbatively, we find that
this fixed point is singly unstable also for sufficiently large $N$
and the critical exponents
show the dimensional reduction.
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2) M. Tissier and G. Tarjus, Phys. Rev. B 74, (2006) 214419. \\
3) Y. Sakamoto, H. Mukaida and C. Itoi, Phys. Rev. B 72 (2005) 144405. \\
4) Y. Sakamoto, H. Mukaida and C. Itoi, Phys. Rev. B 74 (2006) 064402.
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