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 two-dimensional 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 $q\ensuremath{\le}0.5$. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the $0.5<q\ensuremath{\le}1.0$ 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 $\ensuremath{\alpha}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$ depend on the entropic index $q$ in the range $0.5<q\ensuremath{\le}1.0$ in the form $\ensuremath{\alpha}(q)=(10{q}^{2}\ensuremath{-}33q+23)/20$, $\ensuremath{\beta}(q)=(2q\ensuremath{-}1)/8$, and $\ensuremath{\gamma}(q)=({q}^{2}\ensuremath{-}q+7)/4$. On the other hand, the critical exponent $\ensuremath{\nu}$ does not depend on $q$. This suggests a violation of the scaling relations $2\ensuremath{\beta}+\ensuremath{\gamma}=d\ensuremath{\nu}$ and $\ensuremath{\alpha}+2\ensuremath{\beta}+\ensuremath{\gamma}=2$ and a nonuniversality of the critical exponents along the ferro-paramagnetic frontier.