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arXiv:2602.04313 · Nuclear Physics B

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

Georgii Kalagov, Nikita Lebedev

Published Apr 24, 2026• 0 citations
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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 O (3) universality class, as computed within the same FRG/LPA′ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.

DOI
10.1016/j.nuclphysb.2026.117474

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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 O (3) universality class, as computed within the same FRG/LPA′ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.
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We analyze the critical behavior of isotropic systems with dipole-dipole interaction by renormalization-group methods in fixed space-time dimensions. Working in three-dimensional theory we analytically find three-loop expressions for critical exponents in the limit of dominating dipole-dipole forces. Resummation of the series obtained provides numerical values close to O(3)-theory predictions, justifying the applicability of such a simplified model to systems with strong dipole-dipole interaction.
We present an extension of the functional renormalization group (FRG) framework developed to compute critical probability distributions of the order parameter to momentum-dependent observables. Focusing on the constraint effective action at fixed magnetization for the Ising universality class, we derive its exact flow equations and solve them at the second order of the derivative expansion (DE2). We solve these flow equations numerically for two- and three-dimensional systems, extract universal rate functions and momentum-dependent correlation functions, and benchmark them against Monte Carlo simulations. In three dimensions, we recover the rate function and accurately reproduce the first few Fourier modes of the constrained correlation function and demonstrate the convergence of the method. In two dimensions, the lowest-order approximations such as local potential approximation fail, and it is required to consider at least the DE2 to describe the critical point. Our results are in qualitative agreement with the numerics. We confirm the robustness of the FRG approach for calculating both zero- and finite-momentum critical observables at fixed magnetization.
We study a one-dimensional spin-chain interacting with long-range dipolar forces in the presence of a transverse magnetic field. The system exhibits a quantum critical point belonging to the two-dimensional Ising universality class, whose critical behavior is investigated using the nonperturbative density-matrix and functional renormalization group methods. Interestingly, we find that linear spin-wave theory predicts a magnon dispersion characterized by a logarithmic singularity in the magnon velocity for vanishing momenta, due to the peculiar behavior of dipole-dipole interactions in one dimension. However, in the vicinity of the critical point this logarithmically divergent term is renormalized to zero by the effects of quantum fluctuations, signaling the reemergence of scale invariance, in accordance with the Ising critical scenario.