Monte Carlo study of the evaporation/condensation transition on different Ising lattices

Abstract

In 2002 Biskup et al. [Europhys. Lett. 60, 21 (2002)] sketched a rigorous proof for the behavior of the 2D Ising lattice gas, at a finite volume and a fixed excess δM of particles (spins) above the ambient gas density (spontaneous magnetisation). By identifying a dimensionless parameter Δ(δM) and a universal constant Δ_c, they showed in the limit of large system sizes that for Δ< Δ_c the excess is absorbed in the background (``evaporated'' system), while for Δ> Δ_c a droplet of the dense phase occurs (``condensed'' system). To check the applicability of the analytical results to much smaller, practically accessible system sizes, we performed several Monte Carlo simulations for the 2D Ising model with nearest-neighbour couplings on a square lattice at fixed magnetisation M. Thereby, we measured the largest minority droplet, corresponding to the condensed phase, at various system sizes (L=40, >..., 640). With analytic values for for the spontaneous magnetisation m_0, the susceptibility χand the Wulff interfacial free energy density τ_W for the infinite system, we were able to determine λnumerically in very good agreement with the theoretical prediction. Furthermore, we did simulations for the spin-1/2 Ising model on a triangular lattice and with next-nearest-neighbour couplings on a square lattice. Again, finding a very good agreement with the analytic formula, we demonstrate the universal aspects of the theory with respect to the underlying lattice. For the case of the next-nearest-neighbour model, where τ_W is unknown analytically, we present different methods to obtain it numerically by fitting to the distribution of the magnetisation density P(m).