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

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