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.
In ferromagnets with low critical temperatures, dipolar interactions are the reason for the reduction of the critical dimension above which mean-field behavior prevails. Specifically, it is shown that a three-dimensional uniaxial system with dipole–dipole interactions falls into the universality class of a four-dimensional strictly short-range system, whose behavior is mean field with subtle logarithmic corrections which reflect the nonmean-field behavior that sets in fully below four space dimensions.
The behaviour of a d=4- epsilon dimensional model with short-range and weak dipolar interaction is considered near its critical point. The introduction of a dominant interaction between m<or=d 'spin' components allows the discussion of various models including the uniaxial case m=1. Using a paraquat-graph approach it is shown that for m<or= epsilon the increase of both the transverse and the longitudinal spin fluctuations in the m-dimensional subspace contribute to the critical behaviour leading to the peculiar behaviour for m=1. The effective and subcritical exponents are calculated to O( epsilon ) and O(m 0 ). It is found that gamma eff exhibits a pronounced minimum between its asymptotic dipolar and short-range-only values.
We investigate the static critical behaviour of a uniaxial magnetic layer, with finite thickness L in one direction, yet infinitely extended in the remaining d dimensions. The magnetic dipole-dipole interaction is taken into account. We apply a variant of Wilson's momentum shell renormalization group approach to describe the crossover between the critical behaviour of the 3D Ising, 2D Ising, 3D uniaxial dipolar, and the 2D uniaxial dipolar universality classes. The corresponding renormalization group fixed points are in addition to different effective dimensionalities characterized by distinct analytic structures of the propagator, and are consequently associated with varying upper critical dimensions. While the limiting cases can be discussed by means of dimensional expansions with respect to the appropriate upper critical dimensions, respectively, the crossover features must be addressed in terms of the renormalization group flow trajectories at fixed dimensionality d .