Sobre o uso dos zeros da função de partição no estudo de transições de fase

Abstract

Zeros of the partition function were introduced by Fisher as an alternative way to study phase transitions. However, this approach has arisen some issues, the analytical treatment is diffcult and often impossible, requiring numerical methods. Despite the technologicaldevelopment, we still face two major problems for the Fisher zeros numerical solution, the polynomial generated by the canonical partition function has the coeffcient given by density of states and their degree depends on the size of the system. Then we have to deal with high degree polynomials at the order of 30.000 and the imprecision generated by numerical computation using the density of states, which has the difference between their maximal and minimal value at the order of 10400. Aiming the solution to those problems, an alternative method has been proposed [1] based on the Fisher zeros and able to solve these issues. After a simple transformation , the polynomials coeffcients are given by the energy of the system, which has values between [0, 1], if correctly normalized. This way, we have a clear criterion to select the most relevant coeffcients of the polynomial, drastically reducing their degree. In this work, we apply this new method to Ising model and two-dimensional XY, analyzing the effects of the chosen cut in the histogram anddiscussing the behaviour of the proposal algorithm in [1].