Study of the log-periodic oscillations of the specific heat of Cantor energy spectra

Energy spectra with fractal structure are known to lead to a specific heat with log-periodic oscillations as a function of temperature. In this paper, we present a systematic study of the properties of these oscillations for both monoscale and multiscale energy spectra obtained from Cantor sets. We obtain how the amplitude of the oscillations depends on the structure of the spectrum. We also find that the amplitude of the oscillations above the specific heat mean value behaves differently to the amplitude of the oscillations below the mean value. The amplitudes of the latter oscillations has a limiting value given by a characteristic dimension of the spectrum. This asymmetry in the amplitudes produces strong non-harmonic behavior of the oscillatory regime when energy spectra with fractal structure characterized by small fractal dimension are considered. In addition, we also study the behavior of the specific heat when the energy spectrum has unity fractal dimension, i.e. corresponding to an energy spectrum without energy gaps, which are more similar to natural fractal energy spectra than usual Cantor sets. In this case we also find oscillatory behavior of the specific heat if certain conditions are satisfied.