Zeros da função geradora dos momentos: uma nova abordagem para os zeros da função de partição

Abstract

The method of Fisher zeros is used to identify phase transitions without the need to estimate thermodynamic quantities at various temperatures or define an order parameter. However, this method is difficult to apply in practice because it requires solving a highdegree polynomial with coefficients given by the density of states. Furthermore, the values of the density of states and the degree of the polynomial both increase rapidly with system size. These conditions imply that even state-of-the-art root finder algorithms suffer from numerical instabilities, especially for large system sizes. Aiming to solve the Fisher zeros problems, the zeros of the energy probability distribution (EPD) was created. In fact, it is known that the EPD zeros reproduce the results of the Fisher zeros but without some of its problems. The EPD zeros still have one problem, that is, the fast growth of the degree of the polynomial with the system size. To alleviate this problem and to expand the method’s applicability, we proposed a new method that uses the moment-generating function zeros (MGF). It is easy to show that the MGF zeros contain the same information as the Fisher zeros, but with the benefit of a polynomial that is simpler to solve, compared to the EPD zeros method. In addition to that, the MGF zeros polynomial has a low degree that increases slowly as the system size grows. Therefore, this new method is more suitable to be used than the Fisher zeros and the EPD zeros. In this dissertation, we show in detail the development of the MGF zeros method, its relation with the energy probability distribution zeros (EPD), and the relation with a cumulant method. Moreover, using the six-state Potts model and the Ising model in 2 and 3 dimensions, we showed that the MGF zeros yield results statistically equivalent to those of the EPD zeros and the cumulant method. However, when compared to the EPD zeros, the MGF zeros are shown to be computationally cheaper and faster, especially in systems that undergo discontinuous phase transition or that have big lattice sizes. Furthermore, when compared to the cumulant method, the MGF zeros have the advantage of finding more estimates for the critical exponent. Thus, the MGF zeros are an important advance over the partition function zeros.