The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.

Abstract. We studied the pure and dilute Baxter-Wu (BW) models using the WangLandau (WL) sampling method to calculate the Density-Of-States (DOS). We first used the exact result for the DOS of the Ising model to test our code. Then we calculated the DOS of the dilute Ising model to obtain a phase diagram, in good agreement with previous studies. We calculated the energy distribution, together with its first, second and fourth moments, to give the specific heat and the energy fourth order cumulant, better known as the Binder parameter, for the pure BW model. For small samples, the energy distribution displayed a doubly peaked shape, and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with L. The energy distribution yielded the expected BW α = 2/3 critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent θ B equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term θ 1 = 0.248 ± 0.025. For the dilute BW model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected α = 0 was recovered.

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We study the q-state Potts model with four-site interaction on a square lattice. Based on the asymptotic behavior of lattice animals, it is argued that when q≤4 the system exhibits a second-order phase transition and when q>4 the transition is first order. The q=4 model is borderline. We find 1/lnq to be an upper bound on T_{c}, the exact critical temperature. Using a low-temperature expansion, we show that 1/(θlnq), where θ>1 is a q-dependent geometrical term, is an improved upper bound on T_{c}. In fact, our findings support T_{c}=1/(θlnq). This expression is used to estimate the finite correlation length in first-order transition systems. These results can be extended to other lattices. Our theoretical predictions are confirmed numerically by an extensive study of the four-site interaction model using the Wang-Landau entropic sampling method for q=3,4,5. In particular, the q=4 model shows an ambiguous finite-size pseudocritical behavior.