Extended Capillary Waves and the Negative Rigidity Coefficient in the d=2 SOS model

The solid-on-solid (SOS) model of an interface separating two phases is exactly soluble in two dimensions (d=2) when the interface becomes a one-dimensional string. The exact solution in terms of the transfer matrix is recalled and the density-density correlation function $H(z_1,z_2;\mathrm{\Delta }x)$ together with its projections, is computed. It is demonstrated that the shape fluctuations follow the (extended) capillary-wave theory expression $S(q)=kT\mathrm{/}(D+\gamma q^2+\kappa q^4)$ for sufficiently small wave vectors $q$. We find $\kappa$ {\it negative}, $\kappa <0$ . At $q=2\pi$ there is a strong nearest-neighbor peak. Both these results confirm the earlier findings as established in simulations in d=3 and in continuous space, but now in an exactly soluble lattice model.