Extrapolated renormalization-group calculation of the surface tension in square-lattice Ising model

By using self-dual clusters (whose sizes are characterized by the numbers $b=2,3,4,5$), we calculate within a real-space renormalization-group framework, the longitudinal surface tension of the square-lattice first-neighbor ½-spin ferromagnetic Ising model. The exact critical temperature $T_c$ is recovered for any value of $b$; the exact asymptotic behavior of the surface tension in the limit of low temperatures is analytically recovered; the approximate correlation-length critical exponents monotonically tend towards the exact value $\nu =1\mathrm{}$ (which, for two dimensions, coincides with the surface-tension critical exponent $\mu$) for increasingly large cells; the same behavior is noticed in the approximate values for the surface-tension amplitude in the limit $T\to T_c$. We develop four different numerical procedures for extrapolating to $b\to \infty$ the renormalization-group (RG) results for the surface tension, and quite satisfactory agreement is obtained with Onsager's exact expression (error varying from zero to a few percent on the whole temperature domain). Furthermore, we compare the set of RG surface tensions with a set of biased surface tensions (associated with appropriate misfit seams) and find only fortuitous coincidence among them.