Equation of state and scaling relations for isotropic ferromagnets with dipolar interactions

Abstract

The scaling equation of state for ferromagnets with both isotropic exchange and dipolar interactions near the critical point is derived to first order in $\epsilon =4-d$, where $d$ is the dimensionality of space and is also equal to the number of spin components. The resulting scaling function is found to be rather close numerically to its pure short-range counterpart at $\epsilon =1\mathrm{}$, although the functional forms are quite different. Several universal quantities are derived; in particular, the ratio of the amplitudes of the zero-field specific heats above and below $T_c$ is found to be $\frac{A^{+}}{A^{-}}=\frac{6}{5}+O\left(\epsilon \right)$, compared with $\frac{A^{+}}{A^{-}}=1+O\left(\epsilon \right)$ for pure short-range interactions.