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

Georgii Kalagov, Nikita Lebedev

TL;DR

The paper studies the critical behavior of isotropic magnets with strong dipole-dipole interactions in $d=3$, described by a nonconformal Aharony fixed point. It uses a nonperturbative functional renormalization group approach with the LPA$^{\prime}$ truncation to compute the fixed-point potential and the critical exponents $\eta$, $\nu$, and $\omega$ directly in fixed dimension, leveraging a transverse projection that enforces the dipolar constraint. The results show that the dipolar exponents are numerically close to the Heisenberg $O(3)$ values, with $\eta_D \approx 0.0423$, $\nu_D \approx 0.7355$, and $\omega_D \approx 0.7909$, indicating near-degeneracy between the two universality classes and highlighting the need for non-leading universal quantities to distinguish them. The work also discusses regulator dependence and truncation effects, and suggests using nonlinear-susceptibility ratios as additional universal probes to separate the fixed points.

Abstract

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.

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

TL;DR

The paper studies the critical behavior of isotropic magnets with strong dipole-dipole interactions in , described by a nonconformal Aharony fixed point. It uses a nonperturbative functional renormalization group approach with the LPA truncation to compute the fixed-point potential and the critical exponents , , and directly in fixed dimension, leveraging a transverse projection that enforces the dipolar constraint. The results show that the dipolar exponents are numerically close to the Heisenberg values, with , , and , indicating near-degeneracy between the two universality classes and highlighting the need for non-leading universal quantities to distinguish them. The work also discusses regulator dependence and truncation effects, and suggests using nonlinear-susceptibility ratios as additional universal probes to separate the fixed points.

Abstract

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). 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 universality class, as computed within the same FRG/LPA framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.
Paper Structure (15 sections, 51 equations, 2 figures, 1 table)

This paper contains 15 sections, 51 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Effective field model
  3. Functional renormalization group treatment
  4. Flow equation and truncation
  5. Flow of the potential
  6. Wave-function renormalization and $\eta$ exponent
  7. Critical exponents
  8. Conclusions and outlook
  9. Acknowledgements
  10. Propagator in the Wetterich equation
  11. Flow for the running potential
  12. $O(N)$-symmetric model in $d$ spatial demission
  13. Series expansion of the effective potential
  14. The stability matrix
  15. Tensor integrals

Figures (2)

  • Figure 1: Fixed-point potentials $u(\phi)$ at truncation order $N=16$ for the Aharony ($h=\infty$) and Heisenberg ($h=0$) fixed points, shown within the LPA and LPA$^\prime$ truncations.
  • Figure 2: Convergence of the Aharony class critical exponents with the truncation order $N$. Panels show the $N$-dependence of (a) $\nu$, (b) $\omega$, and (c) $\eta$ obtained within the LPA (red triangles) and LPA$^\prime$ (blue circles) truncations. Horizontal lines indicate the values presented in \ref{['tab:exponents']}.