Emergence of log-periodic oscillations in periodic and aperiodic Ising models

This work analyzes the emergence of log-periodic oscillations in thermodynamic functions of Ising models on hierarchical lattices. Several situations, where the exchange interactions are periodic or aperiodic, are taken into account. High precision values for the thermodynamic functions are numerically obtained with the method of transfer matrices. Fitting the curves close to the critical temperature leads to the values of the critical exponents and to the period and amplitude of the oscillations. The first two quantities are found to agree with the results predicted by the renormalization group. The amplitude of oscillations, which are minute for both periodic systems and those with aperiodic irrelevant fluctuations, are significantly enhanced for systems with aperiodic relevant fluctuations. Distinct morphologies of the oscillating pattern are discussed, where oscillations are respectively sinusoidal and with a significant contribution of higher order harmonics.