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Physical review. B, Solid state

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

Amnon Aharony A

Published Oct 1, 1973• 228 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.8.3363

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The exact renormalization-group approach is used to study the critical behavior for T>TcT>{T}_{c}T>Tc​, H=0H=0H=0 of a uniaxial ferromagnetic (or ferroelectric) system in ddd 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 JJJ is the exchange parameter, g\ensuremathμBg{\ensuremath{\mu}}_{B}g\ensuremathμB​ is the magnetic moment per spin, and aaa is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when {t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}, where \ensuremathφ=1+\ensuremathϵ6\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}\ensuremathφ=1+6\ensuremathϵ​ (to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d). For t\ensuremath{\ll}\stackrel{^}{g}, the leading temperature singularity in the Fourier transform of the spin-spin correlation function \ensuremathΓ(q\ensuremath→)\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\ensuremathΓ(q\ensuremath→​) becomes \ensuremathξ2\ifmmode×\else\texttimes\fi[1+(\ensuremathξq)2\ensuremath−h0(\ensuremathξqz)2+g0(qzq)2]\ensuremath−1{\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}\ensuremathξ2\ifmmode×\else\texttimes\fi[1+(\ensuremathξq)2\ensuremath−h0​(\ensuremathξqz)2+g0​(qqz​)2]\ensuremath−1, where h0{h}_{0}h0​ and g0{g}_{0}g0​ are of order \stackrel{^}{g} and \ensuremathξ(t)\ensuremath{\xi}(t)\ensuremathξ(t) varies as t\ensuremath−12{t}^{\ensuremath{-}\frac{1}{2}}t\ensuremath−21​ for d>3d>3d>3, as t\ensuremath−12∣lnt∣16{t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}t\ensuremath−21​∣lnt∣61​ for d=3d=3d=3, and as t\ensuremath−\ensuremathν{t}^{\ensuremath{-}\ensuremath{\nu}}t\ensuremath−\ensuremathν with 12\ensuremathν=1\ensuremath−(3\ensuremath−d)6+O((3\ensuremath−d)2)\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})2\ensuremathν1​=1\ensuremath−6(3\ensuremath−d)​+O((3\ensuremath−d)2) for d<3d<3d<3. The susceptibility displays the expected demagnetization effects, namely, (\ensuremathχ\ensuremath−1\ensuremath−g0D)\ensuremath∝\ensuremathξ\ensuremath−2({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}(\ensuremathχ\ensuremath−1\ensuremath−g0​D)\ensuremath∝\ensuremathξ\ensuremath−2. The experimental situation is mentioned briefly.
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