Using inequalities, we give a simple proof that the surface tension $\tau$ of an Ising model with interaction $\beta J=K$ on a $d$-dimensional lattice, ${\sigma }_i=\pm 1$, satisfies the inequality $\frac{d_{\tau }\mathrm{}\left(K\right)\mathrm{}}{\mathrm{dK}}>~2{\left[m^{*}\mathrm{}\right(K\left)\right]}^2$, where $m^{*}\mathrm{}\left(K\right)\mathrm{}$ is the spontaneous magnetization. When combined with our previous results that $\tau =0\mathrm{}$ for temperatures above $T_{c,s}$, the critical temperature for spontaneous magnetization, this proves that (i) $\tau =0\mathrm{}$, for $T>\mathrm{}{T}_c$, and $\tau >0\mathrm{}$, for $T<\mathrm{}{T}_c$, the critical temperature for the full $d$-dimensional system; and (ii) $T_{c,s}=T_c$. (Both results were known only for $d=2\mathrm{}$).

• Received 19 January 1981

DOI:https://doi.org/10.1103/PhysRevLett.46.1031