Critical behavior of isotropic systems with strong dipole-dipole interaction from the functional renormalization group

We compute the critical exponents of three-dimensional magnets with strong dipole-dipole interactions using the functional renormalization group (FRG) within the local potential approximation including the wave function renormalization (LPA$^\prime$). The system is governed by the Aharony fixed point, which is scale-invariant but lacks conformal invariance. Our nonperturbative FRG analysis identifies this fixed point and determines its scaling behavior. The resulting critical exponents are found to be close to those of the Heisenberg $O(3)$ universality class, as computed within the same FRG/LPA$^\prime$ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.

💡 Research Summary

The paper investigates the critical behavior of three‑dimensional isotropic magnets in which long‑range dipole‑dipole interactions dominate. While short‑range exchange interactions are described by the familiar O(N)‑symmetric φ⁴ theory, the presence of a strong dipolar term modifies the universality class. Historically, Fisher and Aharony showed, using Wilson’s renormalization‑group (RG) in d = 4 − ε dimensions, that a new fixed point—now called the Aharony or dipolar fixed point—appears. This fixed point is scale invariant but does not enjoy full conformal invariance, a fact that renders conformal‑bootstrap techniques inapplicable. Existing perturbative results rely on ε‑expansions up to three loops and on Borel‑resummed series, which, although precise for the Heisenberg O(3) class, remain limited for the dipolar case.

To obtain independent, non‑perturbative estimates, the authors employ the functional renormalization group (FRG) based on the Wetterich flow equation. They work within the leading order of the derivative expansion (the local potential approximation, LPA) but augment it with a running wave‑function renormalization constant Zₖ, a truncation commonly denoted LPA′. The effective average action is taken as

Γₖ