Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering

Renormalisation theory can be used to calculate the statistical properties of systems with arbitrary finite size. This method is demonstrated with the order parameter is and specific heat for epitaxial ordering. The system is renormalised until the unit length is of the order of the total length of the original system. Then, the renormalised partition function is directly evaluated with an appropriate distribution of boundary conditions. Resulting curves of the rounded transition are compared with experimental data. Underlying the finite-size calculation, Migdal-Kadanoff-type recursion relations for q-state Potts models are adjusted by varying q to yield the expected specific-heat exponent in the infinite-system limit. Additionally, it is noted that, with no adjustment, these recursion relations are self-consistently correct for q to infinity on the triangular lattice, and give the first-order transition at the exact temperature and with probably the exact latent heat.