Value ofηfor Long-Range Interactions

The relationship $2-\eta =m$ is derived for the Ising model with a potential of the large-$r$ form $\nu \left(\overrightarrow{\mathrm{r}}\right)\propto -r^{-d-\sigma }$, when $\frac{m}{d}<\frac{1}{2}$ [and for continuum fluids with such a $\nu \left(\overrightarrow{\mathrm{r}}\right)$ when $\frac{m}{d}<\frac{1}{3}$], where $\eta$ has its usual meaning as a critical exponent and $m=min[\sigma ,2]$. On the basis of this result, it is argued that the numerical estimates of Nagle and Bonner indicate a breakdown of strong scaling in the Ising chain with long-range potential.