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Physical review. B, Solid state

Equation of state and scaling relations for isotropic ferromagnets with dipolar interactions

Amnon Aharony A, Alastair D. Bruce

Published Oct 1, 1974• 34 citations
Publisher

Abstract

The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where $d$ is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\ensuremath{\epsilon}=1$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below ${T}_{c}$ is found to be $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})$, compared with $\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})$ for pure short-range interactions.

DOI
10.1103/physrevb.10.2973

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Critical exponents of ferromagnets with dipolar interactions: Second-order ε expansion

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The zero-field critical behavior of ferromagnets above ${T}_{c}$, with both isotropic exchange coupling and dipolar interactions, is studied by renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n (=d)$ component spins. The critical exponents are calculated to order ${\ensuremath{\epsilon}}^{2}$, and are found to be numerically very close to their nondipolar analogs. In particular the correlation length exponent is given by $2\ensuremath{\nu}=1+(\frac{9}{34})\ensuremath{\epsilon}+(\frac{7013}{58956}){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$. The value of the specific-heat exponent ${\ensuremath{\alpha}}_{s} (=2\ensuremath{-}d\ensuremath{\nu})$ seems inconsistent with experimental data (at $d=3$, $\ensuremath{\epsilon}=1$). The critical scattering function is shown to have the form ${\ensuremath{\Gamma}}^{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})=C{t}^{\ensuremath{-}\ensuremath{\gamma}}\stackrel{^}{D}({\ensuremath{\xi}}^{2}{q}^{2})({\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$ to all orders in $\ensuremath{\epsilon}$. The exponents related to the leading correction to scaling, to crossovers to anisotropic exchange behavior, and to cubic dipolar behavior, are also calculated, and are found to differ significantly from their nondipolar counterparts.

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Critical Behavior of Magnets with Dipolar Interactions. I. Renormalization Group near Four Dimensions

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The exact renormalization-group approach of Wilson is used to study the critical behavior for $T>{T}_{c}$, $H=0$, and small $\ensuremath{\epsilon}>0$, of an isotropic ferromagnetic system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, with exchange and dipolar interactions between $d$-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, $J$ is the short-range exchange parameter, and $a$ 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 $n d$ 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.

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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.

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Phase Transitions in Long-Range Ferromagnetic Chains

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Coexistence of dipolar frustration and criticality in ferromagnets

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In real magnets the tendency toward ferromagnetism---promoted by exchange coupling---is usually frustrated by dipolar interaction. As a result, the uniformly ordered phase is replaced by modulated (multidomain) phases, characterized by different order parameters rather than the global magnetization. The transitions occurring within those modulated phases and toward the disordered phase are generally not of second-order type. Nevertheless, strong experimental evidence indicates that a standard critical behavior is recovered when comparatively small fields are applied that stabilize the uniform phase. The resulting power laws are observed with respect to a putative critical point that falls in the portion of the phase diagram occupied by modulated phases, in line with an avoided-criticality scenario. Here we propose a generalization of the scaling hypothesis for ferromagnets, which explains this observation assuming that the dipolar interaction acts as a relevant field, in the sense of renormalization group. We corroborate this proposal with analytic and numerical calculations on the 2D Ising model frustrated by dipolar interaction (isotropic part). Our analysis is directly applicable to thin magnetic films with out-of-plane anisotropy.

arXiv:1806.06046•Physical review. E•Dec 20, 2018•1 citation
The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d, where ddd is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at \ensuremathϵ=1\ensuremath{\epsilon}=1\ensuremathϵ=1, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below Tc{T}_{c}Tc​ is found to be A+A\ensuremath−=65+O(\ensuremathϵ)\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=\frac{6}{5}+O(\ensuremath{\epsilon})A\ensuremath−A+​=56​+O(\ensuremathϵ), compared with A+A\ensuremath−=1+O(\ensuremathϵ)\frac{{A}^{+}}{{A}^{\ensuremath{-}}}=1+O(\ensuremath{\epsilon})A\ensuremath−A+​=1+O(\ensuremathϵ) for pure short-range interactions.
The zero-field critical behavior of ferromagnets above Tc{T}_{c}Tc​, with both isotropic exchange coupling and dipolar interactions, is studied by renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with n(=d)n (=d)n(=d) component spins. The critical exponents are calculated to order ${\ensuremath{\epsilon}}^{2}$, and are found to be numerically very close to their nondipolar analogs. In particular the correlation length exponent is given by $2\ensuremath{\nu}=1+(\frac{9}{34})\ensuremath{\epsilon}+(\frac{7013}{58956}){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$. The value of the specific-heat exponent ${\ensuremath{\alpha}}_{s} (=2\ensuremath{-}d\ensuremath{\nu})$ seems inconsistent with experimental data (at d=3d=3d=3, $\ensuremath{\epsilon}=1$). The critical scattering function is shown to have the form ${\ensuremath{\Gamma}}^{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})=C{t}^{\ensuremath{-}\ensuremath{\gamma}}\stackrel{^}{D}({\ensuremath{\xi}}^{2}{q}^{2})({\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$ to all orders in $\ensuremath{\epsilon}$. The exponents related to the leading correction to scaling, to crossovers to anisotropic exchange behavior, and to cubic dipolar behavior, are also calculated, and are found to differ significantly from their nondipolar counterparts.
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.
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.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between nnn-component spins is investigated near the critical range of ${r}^{\ensuremath{-}2}$. Different formulations of the renormalization group are used for n>1n>1n>1 and n=1n=1n=1. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.
In real magnets the tendency toward ferromagnetism---promoted by exchange coupling---is usually frustrated by dipolar interaction. As a result, the uniformly ordered phase is replaced by modulated (multidomain) phases, characterized by different order parameters rather than the global magnetization. The transitions occurring within those modulated phases and toward the disordered phase are generally not of second-order type. Nevertheless, strong experimental evidence indicates that a standard critical behavior is recovered when comparatively small fields are applied that stabilize the uniform phase. The resulting power laws are observed with respect to a putative critical point that falls in the portion of the phase diagram occupied by modulated phases, in line with an avoided-criticality scenario. Here we propose a generalization of the scaling hypothesis for ferromagnets, which explains this observation assuming that the dipolar interaction acts as a relevant field, in the sense of renormalization group. We corroborate this proposal with analytic and numerical calculations on the 2D Ising model frustrated by dipolar interaction (isotropic part). Our analysis is directly applicable to thin magnetic films with out-of-plane anisotropy.