Critical behavior of the ideal-gas Bose-Einstein condensation in the Apollonian network

We show that the ideal Boson gas displays a finite-temperature Bose-Einstein condensation transition in the complex Apollonian network exhibiting scale-free, small-world, and hierarchical properties. The single- particle tight-binding Hamiltonian with properly rescaled hopping amplitudes has a fractal-like energy spectrum. The energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation. A finite-size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise estimates for the exponents characterizing the condensed fraction, correlation size, and specific heat. The critical exponents, as well as the power-law behavior of the density of states at the bottom of the band, are similar to those of the ideal Boson gas in lattices with spectral dimension ds = 2ln(3)/ ln(9/5) 3.74.