Phase diagrams of semi-infinite systems by renormalization group theory and Monte Carlo simulation

In this work, we used the Migdal-Kadanoff renormalization group method to investigate two three-dimensional semi-infinite Ising systems. In the first one, the surface and bulk sites are occupied by spin-3/2 and spin-1/2 respectively, whereas those of the second one are occupied by spin-1/2 on the surface and spin-3/2 in the bulk. Based on the ratio of bulk and surface exchange interactions, we explored different topologies of phase diagrams showing various second order phase transitions, namely ordinary, extraordinary, surface and special transitions. We also found that both systems exhibited first order phase transitions, multicritical points and critical end-points. The existence of a first order phase transition at low temperatures was confirmed by plotting the derivative of the free energy. The Monte Carlo simulation was also used to verify and compare the results obtained by the renormalization group for the two semi-infinite systems.