Physical review. B, Solid state

Critical Behavior of the Anisotropic n -Vector Model

Michael Ellis Fisher, Pierre M. Pfeuty

Published Sep 1, 1972• 169 citations

Abstract

The critical behavior of a system of $n$-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For $n=3$ and $d=3$ series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).

DOI: 10.1103/physrevb.6.1889

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Crossover Exponent of the Spin Anisotropic n-Vector Model with Short-Range Interaction in 1/n Expansion

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Physical review. B, Solid state

Critical Behavior of the Anisotropic n -Vector Model

Michael Ellis Fisher, Pierre M. Pfeuty

Published Sep 1, 1972• 169 citations

Publisher

Abstract

The critical behavior of a system of

n

-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension

d=4\ensuremath{-}\ensuremath{\epsilon}

. To first order in

\ensuremath{\epsilon}

the correlation and susceptibility exponents in the isotropic limit are

2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}

, while the anisotropy or crossover exponent is

\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}

. When

n\ensuremath{\rightarrow}\ensuremath{\infty}

these expansions agree with exact spherical-model results. For

n=3

and

d=3

series expansions indicate

\ensuremath{\varphi}\ensuremath{\simeq}1.2

(compared with

\ensuremath{\gamma}\ensuremath{\simeq}1.38

DOI: 10.1103/physrevb.6.1889

Related papers

Effects of surface exchange anisotropies on magnetic critical and multicritical behavior at surfaces

Hans Werner Diehl, Erich Eisenriegler

The critical behavior of a semi-infinite $n$ -vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with ${m}_{e}$ (easy-magnetization) components take a critical value above which ${m}_{e}$ -component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis ( ${m}_{e}=1$ ) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.The critical behavior of a semi-infinite $n$-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with ${m}_{e}$ (easy-magnetization) components take a critical value above which ${m}_{e}$-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (${m}_{e}=1$) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.

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

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Critical Behavior of Anisotropic Cubic Systems

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Physical review. B, Solid stateNov 1, 1973233 citations

Critical Behavior of the Anisotropic Classicaln-Vector Model

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Progress of Theoretical PhysicsMay 1, 197312 citations

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S. Hikami, R. Abe

Progress of Theoretical PhysicsAug 1, 197424 citations

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

Alastair D. Bruce, Amnon Aharony A

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.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.

Physical review. B, Solid stateSep 1, 1974122 citations

The critical behavior of a system of

n

-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension

d=4\ensuremath{-}\ensuremath{\epsilon}

. To first order in

\ensuremath{\epsilon}

the correlation and susceptibility exponents in the isotropic limit are

2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}

, while the anisotropy or crossover exponent is

\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}

. When

n\ensuremath{\rightarrow}\ensuremath{\infty}

these expansions agree with exact spherical-model results. For

n=3

and

d=3

series expansions indicate

\ensuremath{\varphi}\ensuremath{\simeq}1.2

(compared with

\ensuremath{\gamma}\ensuremath{\simeq}1.38

The critical behavior of a semi-infinite

n

-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with

{m}_{e}

(easy-magnetization) components take a critical value above which

{m}_{e}

-component surface order is possible while the interaction constants of the remaining ${m}_{h}=n\ensuremath{-}{m}_{e}$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis (

{m}_{e}=1

) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\ensuremath{\epsilon}$.

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 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.