Mean-field calculation of critical parameters and log-periodic characterization of an aperiodic-modulated model

We employ a mean-field approximation to study the Ising model with aperiodic modulation of its interactions in one spatial direction. Two different values for the exchange constant, $J_A$ and $J_B$, are present, according to the Fibonacci sequence. We calculated the pseudo-critical temperatures for finite systems and extrapolate them to the thermodynamic limit. We explicitly obtain the exponents $β$, $δ$, and $γ$ and, from the usual scaling relations for anisotropic models at the upper critical dimension (assumed to be 4 for the model we treat), we calculate $α$, $ν$, $ν_{//}$, $η$, and $η_{//}$. Within the framework of a renormalization-group approach, the Fibonacci sequence is a marginal one and we obtain exponents which depend on the ratio $r \equiv J_B/J_A$, as expected. But the scaling relation $γ= β(δ-1)$ is obeyed for all values of $r$ we studied. We characterize some thermodynamic functions as log-periodic functions of their arguments, as expected for aperiodic-modulated models, and obtain precise values for the exponents from this characterization.

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