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arXiv:2309.02514 · Physical review. B./Physical review. B

Scale without conformal invariance in dipolar ferromagnets

Aleix Gimenez-Grau, Yu Nakayama, Slava Rychkov

Published Jul 16, 2024• 11 citations
arXivHTMLPDF

Abstract

We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.

DOI
10.1103/physrevb.110.024421

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Critical Behavior of Magnets with Dipolar Interactions

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Studies of magnetic critical behavior using the recursion relation approach will be reviewed. For a Hamiltonian including both (anisotropic) exchange and dipole‐dipole interactions many fixed points arise, each representing distinct asymptotic critical behavior. Various non‐dipolar fixed points, for degree of isotropy m, describe the critical behavior of (rotationally invariant) antiferromagnets, except for additional, slowly decaying terms, which can lead to “effective critical exponents”. In addition one finds a series of “dipolar” fixed points. For m>1, these yield exponents which deviate slightly more from the classical values than those for the non‐dipolar fixed points. When m=1 the dipolar fixed point has classical exponents for dimension d>3, plus logarithmic corrections at d=3, but non‐classical behavior for d<3. The dipolar spin‐spin correlation functions always develop a characteristic angular dependence. Spatially anisotropic interactions, antisymmetric exchange interactions, terms of cubic symmetry, etc. may lead to still further fixed points, and hence to new critical behavior.

Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•7 citations
We revisit critical phenomena in isotropic ferromagnets with strong dipolar interactions. The corresponding renormalization group fixed point, dipolar fixed point, was first studied in 1973 by Aharony and Fisher [Phys. Rev. Lett. 30, 559 (1973]. It is distinct from the Heisenberg fixed point, although the critical exponents are close. On the theoretical side, we discuss scale invariance without conformal invariance realized by this fixed point. We elucidate the nonrenormalization of the virial current due to a shift symmetry, and show that the same mechanism is at work in all other known local fixed points which are scale but not conformal invariant. On the phenomenological side, we discuss the relative strength of dipolar and short-range interactions. In some materials, like the europium compounds, dipolar interactions are strong, and the critical behavior is dipolar. In others, like Fe or Ni, dipolar interactions are weaker, and the Heisenberg critical behavior in a range of temperatures is followed by the dipolar behavior closer to the critical point. Some of these effects have been seen experimentally.
Aharony and Fisher showed that nonlocal dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point called the dipolar fixed point. The nonperturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 403{40}^{3}403 by introducing the local cost function parameterized by a parameter $\ensuremath{\lambda}$ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit $\ensuremath{\lambda}=\ensuremath{\infty}$. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant $\ensuremath{\lambda}=8$ [e.g., $\ensuremath{\nu}=0.601(2){(}_{\ensuremath{-}2}^{+0}){{(}_{\ensuremath{-}4}^{+5})}_{{\ensuremath{\beta}}_{c}}$ in the local Heisenberg-dipolar model while $\ensuremath{\nu}=0.712(1){(}_{\ensuremath{-}0}^{+3}){{(}_{\ensuremath{-}1}^{+1})}_{{\ensuremath{\beta}}_{c}}$ in the Heisenberg model], and the spin correlation function has a feature that it becomes divergence free, implying the lack of conformal invariance.
Aharony and Fisher showed that non-local dipolar effects in magnetism destabilize the Heisenberg fixed point in real ferromagnets, leading to a new fixed point, called the dipolar fixed point. The non-perturbative nature of the new fixed point, however, has not been uncovered for many decades. Inspired by the recent understanding that the dipolar fixed point is scale-invariant but not conformal invariant, we perform the Monte Carlo simulation of the local Heisenberg-dipolar model on the lattice of 40340^3403 by introducing the local cost function parameterized by a parameter λλλ and study its critical exponents, which should become identical to the dipolar fixed point of Aharony and Fisher in the infinite coupling limit λ=∞λ= \inftyλ=∞. We find that the critical exponents become noticeably different from those of the Heisenberg fixed point for a finite coupling constant λ=8λ=8λ=8 (e.g. ν=0.601(2)(−2+0)ν=0.601(2)(^{+0}_{-2})ν=0.601(2)(−2+0​) in the local Heisenberg-dipolar model while ν=0.712(1)(−0+3)ν=0.712(1)(^{+3}_{-0})ν=0.712(1)(−0+3​) in the Heisenberg model), and the spin correlation function has a feature that it becomes divergence-free, implying the lack of conformal invariance.
The exact renormalization-group approach of Wilson is used to study the critical behavior for T>TcT>{T}_{c}T>Tc​, H=0H=0H=0, and small $\ensuremath{\epsilon}>0$, of an isotropic ferromagnetic system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, with exchange and dipolar interactions between ddd-component spins. Normal isotropic Heisenberg behavior with $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{\ensuremath{\epsilon}}{4}$ (to first order in $\ensuremath{\epsilon}$) is retained for $t=(\frac{T}{{T}_{c}})\ensuremath{-}1\ensuremath{\ll}\frac{G}{J{a}^{d}}$, where $G=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{2}$ measures the strength of the dipole-dipole interactions, JJJ is the short-range exchange parameter, and aaa is the lattice spacing. When ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\frac{G}{J{a}^{d}}$, where $\ensuremath{\varphi}\ensuremath{\approx}1+\frac{\ensuremath{\epsilon}}{4}$, crossover occurs to a characteristic dipolar behavior described by a new fixed point of the recursion relations. For $t\ensuremath{\ll}\frac{G}{J{a}^{d}}$ one thus finds $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{9\ensuremath{\epsilon}}{34}$ {and, for spins of ndn dnd components, $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}[\frac{(6n+3)}{2(6n+11)}]\ensuremath{\epsilon}$, which agrees with spherical-model results when $n\ensuremath{\rightarrow}\ensuremath{\infty}$}. In the dipolar regime the spin-correlation function $〈{s}_{q}^{\ensuremath{\alpha}}{s}_{\ensuremath{-}q}^{\ensuremath{\beta}}〉$ has a factor [${\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}(\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$], which suppresses longitudinal spin fluctuations; the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ display the expected demagnetization effects. It is found that dipolar anistropies derving from the lattice structure produce weak instabilities which should be hard to detect although their effects are not fully elucidated. Extensions of the results to nonzero magnetic fields, and to anistropic exchange interactions are indicated; the experimental situation is mentioned briefly.
A dipolar fixed point introduced by Aharony and Fisher is a physical example of interacting scale-invariant but nonconformal field theories. We find that the perturbative critical exponents computed in ε expansions violate the conformal bootstrap bound. We formulate the functional renormalization group equations Wetterich and Polchinski to study the fixed point. We present some results in three dimensions within (uncontrolled) local potential approximations (with or without perturbative anomalous dimensions). Published by the American Physical Society 2024
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.
Studies of magnetic critical behavior using the recursion relation approach will be reviewed. For a Hamiltonian including both (anisotropic) exchange and dipole‐dipole interactions many fixed points arise, each representing distinct asymptotic critical behavior. Various non‐dipolar fixed points, for degree of isotropy m, describe the critical behavior of (rotationally invariant) antiferromagnets, except for additional, slowly decaying terms, which can lead to “effective critical exponents”. In addition one finds a series of “dipolar” fixed points. For m>1, these yield exponents which deviate slightly more from the classical values than those for the non‐dipolar fixed points. When m=1 the dipolar fixed point has classical exponents for dimension d>3, plus logarithmic corrections at d=3, but non‐classical behavior for d<3. The dipolar spin‐spin correlation functions always develop a characteristic angular dependence. Spatially anisotropic interactions, antisymmetric exchange interactions, terms of cubic symmetry, etc. may lead to still further fixed points, and hence to new critical behavior.