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 o f parameters and depends only on the topology of the system), the enti re range of parameter values can be studied with minimal additional ef fort. We calculate the DOS functions for the nearest-neighbor (nn) Isi ng 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 signifi cantly extends previous exact calculations of the partition function i n nonzero field (8 x 8 spins for the square lattice). To recognize fea tures of the DOS functions that correspond to phase transitions, we co mpare them with the DOS functions for the Ising chain and for the Isin g model defined on a Sierpinski gasket. The DOS functions define a sur face with respect to the dimensionless independent energy and magnetiz ation variables; this surface is convex with respect to magnetization in the low-energy region for systems displaying a second-order phase t ransition. On the other hand, for systems for which there is no phase transition, the DOS surfaces are concave. We show that this geometrica l property of the DOS functions is generally related to the existence of phase transitions, thereby providing a graphic tool for exploring v arious features of phase transitions. For each given temperature and f ield, we also define a ''free energy surface'', from which we obtain t he most probable energy and magnetization. We test this method of free energy surfaces on Ising systems with both nearest-neighbor (J(1)) an d 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 p hase transition. We also carry out Monte Carlo simulations and compare these quantitatively with our results for the phase diagram.