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

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 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 the $(d,N)$ 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 a dimensional reduction.

9 pages, 2 figures

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• Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study
• Critical scaling in random-field systems: 2 or 3 independent exponents?
• Fixed points and their stability in the functional renormalization group of random field models
• Random field and random anisotropy O(N) spin systems with a free surface
• Comment on ``Random-Field Spin Model beyond 1 Loop: A Mechanism for Decreasing the Lower Critical Dimension"
• One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O($N$) spin model near lower critical dimension
• Critical Behavior of the Three-Dimensional Random Anisotropy Heisenberg Model
• On the breakdown of dimensional reduction and supersymmetry in random-field models