The exact renormalization-group approach of Wilson is used to study the critical behavior for
T>Tc,
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
nd 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 is used to study the critical behavior for
T>Tc,
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
h0 and
g0 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.
By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.
The purpose of this article is to give an overview of results concerning ordering and critical properties of two-dimensional ferromagnets including the dipolar interaction.We investigate a two-dimensional xy-model extended by the dipolar interaction.Describing our system by a nonlinear σ-model and using renormalization group methods we predict a phase transition to an ordered state.This transition is due to the long-range dipolar interaction.The ferromagnetic phase is governed by a low temperature fixed-point with infinite dipolar coupling.In the critical regime we find exponential behavior for the correlation length and the order parameter in contrast to the usual power laws.The nature of the transition shows a striking similarity to the Kosterlitz-Thouless transition.We show that there is a whole class of longrange xy-models leading to such non-standard behavior.Parameterizing the divergencies in terms of the correlation length we are able to calculate the critical exponents.These exponents are correct in any loop order.
In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width
n (hn) and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants $\ensuremath{\delta}$. The obtained results indicate that the
h1-TL phase transition line is continuous up to $\ensuremath{\delta}=1.2585$, while for the
h2-TL line a weak first-order character is found in the interval $1.2585\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.36$ and becomes continuous for $1.37\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1.9$. This result suggests the existence of a tricritical point close to $\ensuremath{\delta}=1.37$. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on $\ensuremath{\delta}$ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the ${h}_{1}\ensuremath{-}{h}_{2}$ line, i.e., at $\ensuremath{\delta}=1.2585$, the critical phase corresponding to the ${h}_{1}\ensuremath{-}\mathrm{TL}$ transition coexists with the
h2 phase.