Critical Behavior of Magnets with Dipolar Interactions. II. Feynman-Graph Expansion for Ferromagnets near Four Dimensions

Abstract

The Feynman-graph-expansion approach of Wilson is used to study the critical behavior for $0<t=\left(\frac{T}{T_c}\right)-1\mathrm{}\ll \frac{G}{J{a}^d}$ and $H=0\mathrm{}$ of an isotropic ferromagnet in $d=4-\epsilon (\epsilon >0)$ dimensions with exchange and dipolar interactions between $d$-component spins. [Here $G=\frac{{\left(g{\mu }_B\right)}^2}{2}$ measures the dipole-dipole interaction strength, $J$ is the short-range exchange parameter, and $a$ is the lattice spacing.] The susceptibility and the two-spin correlation functions are calculated to first order in $\epsilon$, and agree with previous work, based on the renormalization-group approach. In addition the correlation function for transverse-spin fluctuations at $T_c$ is investigated, yielding the critical exponent $\eta \approx \frac{20{\epsilon }^2}{867}$ [whereas for short-range exchange forces one has $\eta \approx \frac{3{\epsilon }^2}{{12}^2}$]. The limiting angular dependence of the four-spin correlation function is obtained.