Area constrained SOS models of interfaces

The solid-on solid (SOS) model in two dimensions ($d=2$) is now solved under the constraint of constant energy and then under the new constraint of constant total area. From the combinatorial factors $g(E;L,M)$, the new ensemble is constructed with its free energy $F(A_{tot},T)$ of a membrane of constant (onedimensional) area $A_{tot}$. The entropy per column $Y=(1/L)\log g(E;L,M)$ of rectangular $L\times M$ strips reduces to a common curve in reduced variables. Definitions of the "area" and of the interfacial tension, are compared or discussed. Analytical calculations are supported with numerical ones and vice versa. Overall the constraint reduces the ratio of $A_{tot}$ to the projected area $L$, as compared with canonical calculation, strongly at high temperatures.