Exactly solvable tight-binding model on the RAN: fractal energy spectrum and Bose–Einstein condensation

The Regularized Apollonian Network (RAN) is defined starting from a tetrahedral structure with four nodes all connected (generation 0). At any successive generations, new nodes are added and connected with the surrounding three nodes. As a result, a power-law cumulative distribution of connectivity P(k) ∝ 1/k η with η = ln(3)/ln(2) ≈ 1.585 is obtained. We initially consider a single- particle tight-binding model on the RAN. The eigenvalues of the Hamiltonian are exactly computed by a recursive approach for any size of the network. In the infinite size limit, the density of states and the cumulative distribution of states (integrated density of states) are also exactly determined. The relevant scaling behavior of the cumulative distribution close to the band bottom is shown to be a power law with an exponent depending on the spectral dimension and not on the dimension of the embedding space. We then consider a gas made by an infinite number of non-interacting bosons, each of them described by the tight-binding Hamiltonian on the RAN and we prove that, for sufficiently large bosonic density and sufficiently small temperature, a macroscopic fraction of the particles occupies the lowest single-particle energy state forming the Bose–Einstein condensate. Not only do we determine the transition temperature as a function of the bosonic density, but also the fraction of condensed particles, the fugacity, the energy and the specific heat for any temperature and bosonic density.