Bosons with multifractal energy spectrum: specific heat log periodicity and Bose–Einstein condensation

Abstract

The thermodynamics for systems of non-interacting bosons with multifractal energy spectrum is considered. The critical attractors of one-dimensional generalized logistic and circular maps are used to generate multifractal bounded spectra with well defined scaling exponents. The specific heat is then calculated for both cases of conserved and non-conserved particle number, showing a power-law behaviour which is modulated by log-periodic oscillations when the energy spectrum is not dense. The occurrence of Bose–Einstein condensation for systems with conserved particle number, at which the specific heat is discontinuous, is also analyzed.

Specific heat spectra for quasiperiodic ladder sequences

“…Simplified fractals based in the Cantor [6,7], as well as the critical attractor of the logistic and circle maps at the onset of chaos [8][9][10], have been used recently to model the energy spectrum of quasiperiodic systems. The thermodynamic behavior derived from such self-similar spectra display some anomalous features, with the most prominent one being related to the emergence of log-periodic oscillations in the low-temperature behavior of the specific heat.…”

Observation of log-periodic oscillations in the quantum dynamics of electrons on the one-dimensional Fibonacci quasicrystal

“…As such, they naturally occur in quasiperiodic systems like the Fibonacci quasicrystal, where the spectrum of energies is multifractal. This has been observed in quasiperiodic Ising models [32], and in specific heat studies of the tight-binding Fibonacci model [33][34][35][36][37].…”