Specific heat anomalies associated with Cantor-set energy spectra

Most physical models on quasicrystals, as well as the related experimental results, exhibit fractal energy spectra. In order to have a deep insight on relevant thermodynamic implications of this feature, we have performed analytical and high precision numerical calculations of the specific heats Cₙᵇᵃⁿᵈ and Cₙᵈⁱˢᶜ associated with successive hierarchical approximations (n=1,2,3,) to bounded Cantor-set energy spectra (constructed with sets of continuous intervals for the banded case, and with discrete levels for the discretecase). Instructive anomalies are exhibited, namely (i) Cₙᵇᵃⁿᵈ(T) and Cₙᵈⁱˢᶜ(T) differ for all temperatures and finite n (in particular, in units of kB, Cₙᵈⁱˢᶜ(0)=0 whereas Cₙᵇᵃⁿᵈ(0)=1), but, through an interesting nonuniform convergence, C_∞ᵇᵃⁿᵈ(T)=C_∞ᵈⁱˢᶜ(T)≡C_∞(T) for all finite temperatures; (ii) in the T→0 limit, C_∞(T) exhibits an infinite number of small-amplitude oscillations symmetrically disposed precisely around the fractal dimensionality df=ln2/ln3; more precisely, C_∞(T)~C*(T), where C*(T)=C*(3T)=∑_k=-∞^∞[3ᵏTcosh(1/3ᵏT)]^-2 =ln2/ln3+asin[2πln(bT)/ln3]+ε(T) with a=1.27×10^-2, b=1.97 and ε(T)<5×10^-4 (∀ T) $(T$ is measured in units of the outermost width of the Cantor set); (iii) in the T→∞ limit, C_∞(T)~1/8T². In addition to this, we comment on a possible connection of this type of systems with the recently introduced nonextensive thermostatistics.