The two-point correlation function for a random Ising model at d=4 is obtained using the solutions of the renormalisation-group recursion relations in the range of temperatures where the system exhibits non-asymptotic critical behaviour, the origin of which lies in a particular degeneracy of the RG recursion relations. The effects of lattice compressibility are also studied with two types of boundary conditions, corresponding to constant volume and pressure, respectively. A comparison with pure Ising systems is made.
Thermodynamic functions of an
n-component magnetic system coupled to an isotropic elastic medium in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are calculated to leading order in $\ensuremath{\epsilon}$. Solutions to renormalization-group recursion relations correct to $O(\ensuremath{\epsilon})$ along with trajectory-integral methods are employed in these calculations. For
n> 2 the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the
n=1 case renormalization-group predictions of a first-order transition are verified. Thermodynamic functions are given and analyzed in this case. In particular the transition temperature and discontinuity in order parameter are determined and the phase diagram is constructed.
Renormalization-group equations are exactly solved for the random Ising model with (i) short-range interaction at
d=4, and (ii) dipolar interactions at
d=3. In both cases, the leading singularities of the susceptibility $\ensuremath{\chi}$ and of the specific heat
C are found to be $\ensuremath{\chi}\ensuremath{\propto}{t}^{\ensuremath{-}1}\mathrm{exp}[{(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ and $C\ensuremath{\propto}\ensuremath{-}{|\mathrm{ln}t|}^{\frac{1}{2}}\mathrm{exp}[{\ensuremath{-}2(D|\mathrm{ln}t|)}^{\frac{1}{2}}]$ as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$.
D is a universal constant, equal to 6/53 in case (i) and to 9/[81ln(4/3) + 53] in case (ii). Relations between amplitudes of
C and of the correlation length, corrections to the leading singularities, crossover effects from the nonrandom region or from the meanfield region to the asymptotic critical region and possible experiments are also discussed.
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.
The critical behavior of a compressible Ising system of the Baker-Essam type is examined for positive pressures. In addition to satisfying the assumptions of Fisher's theory of renormalization of critical exponents, the model, allows one to calculate the temperature region around
Tc where the renormalized exponents should show up. The region is $\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}$ for small pressures, where
J1 is the spin-lattice coupling and $\ensuremath{\alpha}$ is the specific-heat exponent of the corresponding rigid Ising system.
The renormalization group approach is applied to the study of the short-time critical behaviour of the d -dimensional n -component anisotropic cubic spin systems with long-range interaction of the form p σ s p s - p in momentum space. Firstly, the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents θ' and θ of the order parameter and the response function, respectively, are calculated to the second order in = 2σ- d . For 1⩽ d <2σ and n > n c , the cubic anisotropy affects the short-time critical behaviour.