Finite size analysis of a two-dimensional Ising model within a nonextensive approach

Abstract

In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic 2D Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for $q\le 0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\le 1.0$ regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents $\alpha$, $\beta$ and $\gamma$ depend on the entropic index $q$ in the range $0.5<q\le 1.0$ in the form $\alpha (q)=(10q^2-33q+23)/20$, $\beta (q)=(2q-1)/8$ and $\gamma (q)=(q^2-q+7)/4$. On the other hand, the critical exponent $\nu$ does not depend on $q$. This suggests a violation of the scaling relations $2\beta +\gamma =d\nu$ and $\alpha +2\beta +\gamma =2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.

accepted for publication in Phys. Rev. E