# 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

## TL;DR

This paper uses one-loop functional renormalization group analysis to study critical phenomena, dimensional reduction, and its breakdown in long-range random field O(N) spin models near the lower critical dimension, revealing stability conditions and boundary behaviors.

## Contribution

It provides the first analytical fixed points and stability analysis for long-range random field O(N) models, clarifying when dimensional reduction holds or breaks down.

## Key findings

- Dimensional reduction holds only for N > 18.3923 in certain long-range models.
- Analytic fixed points can be destabilized by nonanalytic perturbations.
- Boundary between long-range and short-range critical behaviors matches Sak's prediction.

## Abstract

We consider the random-field O($N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as $r^{-d-\sigma}$ and/or random fields which correlate with distance $r$ as $r^{-d+\rho}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\to d-\sigma$ dimensional reduction at the leading order of the $d-2\sigma$ expansion holds only for $N>2(4+3{\sqrt{3}})\simeq 18.3923\cdots$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B {\bf{8}}, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\to d-\sigma-\rho$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.