Geometric criteria for phase transitions: The Ising model with nearest and next-nearest neighbor interactions

We describe a geometric approach for studying phase transitions, based upon the analysis of the ''density of states'' (DOS) functions (exact partition functions) for finite Ising systems. This approach presents a complementary method to the standard Monte Carlo method, since with a single calculation of the density of states (which is independent of parameters and depends only on the topology of the system), the entire range of parameter values can be studied with minimal additional effort. We calculate the DOS functions for the nearest-neighbor (nn) Ising model in nonzero field for square lattices up to 12 x 12 spins, and for triangular lattices up to 12 spins in the base; this work significantly extends previous exact calculations of the partition function in nonzero field (8 x 8 spins for the square lattice). To recognize features of the DOS functions that correspond to phase transitions, we compare them with the DOS functions for the Ising chain and for the Ising model defined on a Sierpinski gasket. The DOS functions define a surface with respect to the dimensionless independent energy and magnetization variables; this surface is convex with respect to magnetization in the low-energy region for systems displaying a second-order phase transition. On the other hand, for systems for which there is no phase transition, the DOS surfaces are concave. We show that this geometrical property of the DOS functions is generally related to the existence of phase transitions, thereby providing a graphic tool for exploring various features of phase transitions. For each given temperature and field, we also define a ''free energy surface'', from which we obtain the most probable energy and magnetization. We test this method of free energy surfaces on Ising systems with both nearest-neighbor (J(1)) and next-nearest-neighbor (J(2)) interactions for various values of the ratio R = J(1)/J(2). For one particular choice, R = -0.1, we show how the ''free energy surface'' may be utilized to discern a first-order phase transition. We also carry out Monte Carlo simulations and compare these quantitatively with our results for the phase diagram.