Physical review. B, Solid state

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

Alastair D. Bruce, Amnon Aharony A

Published Sep 1, 1974• 122 citations

Abstract

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.

DOI: 10.1103/physrevb.10.2078

Related papers

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

Amnon Aharony A, Alastair D. Bruce

Physical review. B, Solid stateOct 1, 197434 citations

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

Amnon Aharony A, Michael Ellis Fisher

Physical review. B, Solid stateOct 1, 1973308 citations

Critical Behavior of Anisotropic Cubic Systems

Amnon Aharony A

Physical review. B, Solid stateNov 1, 1973233 citations

Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in d Dimensions

Amnon Aharony A

Physical review. B, Solid stateOct 1, 1973228 citations

The renormalization group in the theory of critical behavior

Michael Ellis Fisher

Reviews of Modern PhysicsOct 1, 19741,551 citations

Critical properties of random and constrained dipolar magnets

Amnon Aharony A

Physical review. B, Solid stateAug 1, 197540 citations

Physical review. B, Solid state

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

Alastair D. Bruce, Amnon Aharony A

Published Sep 1, 1974• 122 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.10.2078

Related papers

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

Amnon Aharony A, Alastair D. Bruce

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.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.

Physical review. B, Solid stateOct 1, 197434 citations

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

Amnon Aharony A, Michael Ellis Fisher

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.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.

Physical review. B, Solid stateOct 1, 1973308 citations

Critical Behavior of Anisotropic Cubic Systems

Amnon Aharony A

The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$ -component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $n$-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

Physical review. B, Solid stateNov 1, 1973233 citations

Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in d Dimensions

Amnon Aharony A

The exact renormalization-group approach is used to study the critical behavior for $T>{T}_{c}$ , $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and $a$ is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where ${h}_{0}$ and ${g}_{0}$ are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for $d>3$ , as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for $d=3$ , and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for $d<3$ . The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.The exact renormalization-group approach is used to study the critical behavior for $T>{T}_{c}$, $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and $a$ is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where ${h}_{0}$ and ${g}_{0}$ are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for $d>3$, as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for $d=3$, and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for $d<3$. The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.

Physical review. B, Solid stateOct 1, 1973228 citations

The renormalization group in the theory of critical behavior

Michael Ellis Fisher

The renormalization group approach to the theory of critical behavior is reviewed at an introductory level with emphasis on magnetic systems. Among recent results reported are the dependence of critical exponents above ${T}_{c}$ on dimensionality $d=4\ensuremath{-}\ensuremath{\epsilon}$; on the symmetry index or number of spin components, $n$ ; on the range and anisotropy of exchange couplings; and on dipolar interactions and lattice anisotropies, in ferro- and antiferromagnets. Calculations of the scaling functions for the equation of state and critical scattering are summarized.The renormalization group approach to the theory of critical behavior is reviewed at an introductory level with emphasis on magnetic systems. Among recent results reported are the dependence of critical exponents above ${T}_{c}$ on dimensionality $d=4\ensuremath{-}\ensuremath{\epsilon}$; on the symmetry index or number of spin components, $n$; on the range and anisotropy of exchange couplings; and on dipolar interactions and lattice anisotropies, in ferro- and antiferromagnets. Calculations of the scaling functions for the equation of state and critical scattering are summarized.

Reviews of Modern PhysicsOct 1, 19741,551 citations

Critical properties of random and constrained dipolar magnets

Amnon Aharony A

Renormalization-group techniques are used to study a model of $n$ -coupled $m$ -component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.Renormalization-group techniques are used to study a model of $n$-coupled $m$-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.

Physical review. B, Solid stateAug 1, 197540 citations

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}

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.

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.

The critical behavior in zero field above

{T}_{c}

of ferromagnets or ferroelectrics with a Hamiltonian of cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with

n

-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, a crossover from isotropic (Heisenberg) to characteristic cubic behavior occurs, with the new value $2{v}^{C}=1+[\frac{(n\ensuremath{-}1)}{3n}]\ensuremath{\epsilon}+[\frac{(n\ensuremath{-}1)}{324{n}^{3}}](17{n}^{2}+290n\ensuremath{-}424){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, and cubic symmetry appearing in the four-spin correlation function. Experiments on structural phase transitions are considered briefly.

The exact renormalization-group approach is used to study the critical behavior for

T>{T}_{c}

H=0

of a uniaxial ferromagnetic (or ferroelectric) system in

d

dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where

J

is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and

a

is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where

{h}_{0}

and

{g}_{0}

are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for

d>3

, as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for

d=3

, and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for

d<3

. The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.

{T}_{c}

on dimensionality $d=4\ensuremath{-}\ensuremath{\epsilon}$; on the symmetry index or number of spin components,

n

; on the range and anisotropy of exchange couplings; and on dipolar interactions and lattice anisotropies, in ferro- and antiferromagnets. Calculations of the scaling functions for the equation of state and critical scattering are summarized.

Renormalization-group techniques are used to study a model of

n

-coupled

m

-component spin systems, in the limit of large dipole-dipole interactions. As recently shown by Emery, this model describes the critical behavior of a constrained dipolar system in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ and that of a dipolar system with a quenched random perturbation in the limit $n\ensuremath{\rightarrow}0$. In both cases, the unperturbed dipolar fixed point is unstable, and there is a crossover to a new behavior. For the constrained system, this leads to another dipolar fixed point (if $\ensuremath{\alpha}<0$, $\ensuremath{\alpha}$ being the dipolar specific-heat exponent) with the same thermodynamic critical exponents, or to one with renormalized dipolar exponents (if $\ensuremath{\alpha}>0$). These results are different from those of previous "spherical" dipolar models. For the random case, the crossover is either to a new fixed point, with very different exponents, e.g., $2\ensuremath{\nu}\ensuremath{\simeq}1+1.183\ensuremath{\epsilon}$ for $m=d=4\ensuremath{-}\ensuremath{\epsilon}$, or away from all the fixed points found to order $\ensuremath{\epsilon}$. One of the new fixed points in this case has complex eigenvalues of the linearized recursion relations. This is related to the fact that the recursion-relation flow is not of a gradient type for random systems.