Monte Carlo study of the droplet formation-dissolution transition on different two-dimensional lattices - Nuβbaumer, Andreas; Bittner, Elmar; Janke, Wolfhard (Physical review. E, Statistical, nonlinear, and soft matter physics, 2008)
In 2003 Biskup et al. [Commun. Math. Phys. 242, 137 (2003)] gave a rigorous proof for the behavior of equilibrium droplets in the two-dimensional (2D) spin-1/2 Ising model (or, equivalently, a lattice gas of particles) on a finite square lattice of volume V with a given excess δM≡M−M0 of magnetization compared to the spontaneous magnetization M0=m0V. By identifying a dimensionless parameter Δ(δM) and a universal constant Δc, they showed in the limit of large system sizes that for Δ<Δc the excess is absorbed in the background (“evaporated” system), while for Δ>Δc a droplet of the minority phase occurs (“condensed” system). By minimizing the free energy of the system, they derived an explicit formula for the fraction λ(Δ) of excess magnetization forming the droplet. To check the applicability of the asymptotic analytical results to much smaller, practically accessible, system sizes, we performed several Monte Carlo simulations of the 2D Ising model with nearest-neighbor couplings on a square lattice at fixed magnetization M. Thereby, we measured the largest minority droplet, corresponding to the condensed phase in the lattice-gas interpretation, at various system sizes (L=40,80,…,640). With analytical values for the spontaneous magnetization density m0, the susceptibility χ, and the Wulff interfacial free energy density τW for the infinite system, we were able to determine λ numerically 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-neighbor 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 and type of interaction. For the case of the next-nearest-neighbor model, where τW is unknown analytically, we present different methods to obtain it numerically by fitting to the distribution of the magnetization density P(m).