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Physical review. B, Solid state

Quadrupolar interactions at ferromagnetic critical points

Amnon Aharony A

Published Mar 1, 1974• 17 citations
Publisher

Abstract

The critical behavior of $d$-dimensional ferromagnets with quadrupole-quadrupole short- and long-range interactions is discussed on the basis of renormalization-group calculations to orber ${\ensuremath{\epsilon}}^{2}$, with $\ensuremath{\epsilon}=4\ensuremath{-}d$. Such interactions may lead to a first-order transition, to characteristic "cubic" critical behavior, or to the usual isotropic critical behavior.

DOI
10.1103/physrevb.9.2416

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The critical behavior of ddd-dimensional ferromagnets with quadrupole-quadrupole short- and long-range interactions is discussed on the basis of renormalization-group calculations to orber \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2, with \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d. Such interactions may lead to a first-order transition, to characteristic "cubic" critical behavior, or to the usual isotropic critical behavior.
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.
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.
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The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in d=4d=4d=4 dimension.