Critical behavior of the two-dimensional Ising susceptibility

Exploring different regimes in finite-size scaling of the droplet condensation-evaporation transition

We present a finite-size scaling analysis of the droplet condensation-evaporation transition of a lattice gas (in two and three dimensions) and a Lennard-Jones gas (in three dimensions) at fixed density. Parallel multicanonical simulations allow sampling of the required system sizes with precise equilibrium estimates. In the limit of large systems, we verify the theoretical leading-order scaling prediction for both the transition temperature and the finite-size rounding. In addition, we present an emerging intermediate scaling regime, consistent in all considered cases and with similar recent observations for polymer aggregation. While the intermediate regime locally may show a different effective scaling, we show that it is a gradual crossover to the large-system scaling behavior by including empirical higher-order corrections. This implies that care has to be taken when considering scaling ranges, possibly leading to completely wrong predictions for the thermodynamic limit. In this study, we consider a crossing of the phase boundary orthogonal to the usual fixed temperature studies. We show that this is an equivalent approach and, under certain conditions, may show smaller finite-size corrections.

Exact solution and high-temperature series expansion study of the one-fifth-depleted square-lattice Ising model

The critical behavior of the one-fifth-depleted square-lattice Ising model with nearest-neighbor ferromagnetic interaction has been investigated by means of both an exact solution and a high-temperature series expansion study of the zero-field susceptibility. For the exact solution we employ a decoration transformation followed by a mapping to a staggered eight-vertex model. This yields a quartic equation for the critical coupling giving K(c)(≡βJ(c))=0.695. The series expansion for the susceptibility, to O(K(18)), when analyzed via standard Padé approximant methods gives an estimate of K(c), consistent with the exact solution result to at least four significant figures. The series expansion is also analyzed for the leading amplitude and subdominant terms.

Monte Carlo study of the droplet formation-dissolution transition on different two-dimensional lattices

In 2003 Biskup [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 delta M identical with M-M 0 of magnetization compared to the spontaneous magnetization M 0=m0V . By identifying a dimensionless parameter Delta(delta M) and a universal constant Delta c , they showed in the limit of large system sizes that for Delta<Delta c the excess is absorbed in the background ("evaporated" system), while for Delta>Delta c 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 lambda(Delta) 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 m 0 , the susceptibility chi , and the Wulff interfacial free energy density tau W for the infinite system, we were able to determine lambda 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 tau W is unknown analytically, we present different methods to obtain it numerically by fitting to the distribution of the magnetization density P(m) .