Effects of heterogeneity in site–site couplings for tight-binding models on scale-invariant structures

Abstract

We studied the thermodynamic behaviors of non-interacting bosons and fermions trapped by a scale-invariant branching structure of adjustable degree of heterogeneity. The full energy spectrum in tight-binding approximation was analytically solved . We found that the log-periodic oscillation of the specific heat for Fermi gas depended on the heterogeneity of hopping. Also, low dimensional Bose-Einstein condensation occurred only for non-homogeneous setup.

Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network

“…The hierarchical tree we consider here is constructed iteratively in a self-similar pattern. Actually, this structure corresponds to the particular case G(m = 1, θ = 0) of a general tree defined in [34]. Following the introduced definition and notation, G 0 is a line with two nodes.…”

“…yields the eigenvalue ε ′ of the Hamiltonian H t−1 for G t−1 . A more systematic treatment of the flow equation related to a general tree can be found at [34]. Note that S(ε) provides only one half of the hierarchical structure of the full spectrum.…”

“…Actually, this structure corresponds to the particular case G(m = 1, θ = 0) of a general tree defined in [34]. Following the introduced definition and notation, G 0 is a line with two nodes.…”