Universal size dependence of the free energy of finite systems near criticality

A scaling framework is developed for the emergence of the universal finite-size correction $\frac{\left(u\ln L\right)}{L^2}$, recently found by Cardy and Peschel in the free-energy density of the two-dimensional (2D) conformally invariant models at $T_c$. Our approach applies in the critical region $T\simeq T_{\mathrm{cZ}}$, and accounts for the universality of $u$, but not its value. We predict a similar universal term, $\frac{\left(u\ln L\right)}{L^3}$, in 3D. We also find that the "corner" free energy in the bulk ($L=\infty$) has the leading singular behavior ($-\nu u\ln \mathrm{}\left|t\right|\mathrm{}$).