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

TL;DR

This paper uses mean-field theory to analyze an aperiodic-modulated Ising model with Fibonacci sequence interactions, deriving critical exponents, studying log-periodic behavior, and confirming scaling relations.

Contribution

It provides a mean-field analysis of the critical behavior of an aperiodic Ising model with Fibonacci modulation, including the calculation of critical exponents and log-periodic functions.

Findings

Exponents depend on the ratio J_B/J_A.

Scaling relation γ = β(δ - 1) holds for all studied ratios.

Thermodynamic functions exhibit log-periodic oscillations.

Abstract

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 $\beta$, $\delta$, and $\gamma$ 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 $\alpha$, $\nu$, $\nu_{//}$, $\eta$, and $\eta_{//}$. 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 $\gamma = \beta (\delta -1)$ 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.