The solid-on-solid surface width around the roughening transition

We investigate the surface width $W$ of solid-on-solid surfaces in the vicinity of the roughening temperature $T_r$. Above $T_r$, $W^2$ is expected to diverge with the system size $L$ like $\ln L$. However, close to $T_r$ a clean $\ln{L}$ behavior can only be seen on extremely large lattices. Starting from the Kosterlitz-Thouless renormalization group, we derive an improved formula that describes the small $L$ behavior on both sides of $T_r$. For the Discrete Gaussian model, we used the valleys-to-mountains-reflections cluster algorithm in order to simulate the fluctuating solid-on-solid surface. The base plane above which the surface is defined is an $L \times L$ square lattice. In the simulation we took $8\leq L\leq 256$. The improved formula fits the numerical results very well. {}From the analysis, we estimate the roughening temperature to be $T_r = 0.755(3)$.