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.
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 .
The zero-field critical behavior of ferromagnets above
Tc, 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.
Renormalisation group recursion relations and matching techniques are used to evaluate to O( epsilon ) the crossover scaling function describing the competition between isotropic short-range and dipolar critical behaviours in the two-point correlation function of a magnetic system above T c in zero field. The scaling function, obtained in closed form, interpolates between the small-q (q xi <<1) and large-q (q xi >>1) limits where asymptotic expressions are written. The results for the effective critical exponents are compared with neutron scattering measurements.
Scaling and renormalization-group theories of crossover in criticality induced by random interactions are searchingly tested for Ising (n=1) systems. Extrapolation of 15th-order series in two variables for the susceptibility of the \ifmmode\pm\else\textpm\fi{}J simple-cubic Ising model yields a crossover exponent \ensuremath{\varphi}=0.125\ifmmode\pm\else\textpm\fi{}0.025. This agrees well with the specific-heat exponent, \ensuremath{\alpha}\ensuremath{\simeq}0.11. Consistency between free-energy and susceptibility scaling with \ensuremath{\varphi}=\ensuremath{\alpha} is cross-checked and leading universal scaling-function coefficients and nonuniversal scaling field amplitudes are determined.