Effect of second-rank random anisotropy on critical phenomena of random field O(N) spin model in the large N limit

TL;DR

This paper investigates the critical behavior of a large N limit of the random field O(N) spin model with second-rank anisotropy, confirming dimensional reduction and stability of the replica-symmetric solution.

Contribution

It provides a detailed analysis of the stability of the replica-symmetric saddle point and fixed points, demonstrating dimensional reduction holds for large N in this model.

Findings

Replica-symmetric saddle point is stable against fluctuations.

Dimensional reduction holds in the large N limit.

Analytic fixed point is nearly singly unstable, supporting dimensional reduction.

Abstract

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