Exact Relations among Amplitudes at Critical Points of Marginal Dimensionality

A renormalization-group analysis of a three-dimensional Ising model with dipolar forces establishes a relation between the specific heat and the correlation length, in the limit $t=\frac{(T-T_c)}{T_c}\to 0$. Specific-heat data are used to predict $\zeta \sim 1.41t^{-\frac{1}{2}}{\left|\ln t\right|}^{\frac{1}{6}}$ Å for LiTb${\mathrm{F}}_4$. A relation among the magnetization, the specific heat, and the susceptibility is also given. Similar relations are found for $n$-component short-range models at $d=4\mathrm{}$ and $d=4-\epsilon$.