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We study the critical behavior of a random field O($N$) spin model with asecond-rank random anisotropy term in spatial dimensions $4<d<6$, by means ofthe replica method and the 1/N expansion. We obtain a replica-symmetricsolution of the saddle-point equation, and we find the phase transition obeyingdimensional reduction. We study the stability of the replica-symmetric saddlepoint against the fluctuation induced by the second-rank random anisotropy. Weshow that the eigenvalue of the Hessian at the replica-symmetric saddle pointis strictly positive. Therefore, this saddle point is stable and thedimensional reduction holds in the 1/N expansion. To check the consistency withthe functional renormalization group method, we obtain all fixed points of therenormalization group in the large $N$ limit and discuss their stability. Wefind that the analytic fixed point yielding the dimensional reduction ispractically singly unstable in a coupling constant space of the given modelwith large $N$. Thus, we conclude that the dimensional reduction holds forsufficiently large $N$.

physical review. b, condensed matter and materials physics

Effect of second-rank random anisotropy on critical phenomena of a random-field<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>spin model in the large<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>limit

Effect of second-rank random anisotropy on critical phenomena of a random-fieldO(N)spin model in the largeNlimit