Bose-Einstein condensation in diamond hierarchical lattices

The Bose-Einstein condensation of noninteracting particles restricted to move on the sites of hierarchical diamond lattices is investigated. Using a tight-binding single-particle Hamiltonian with properly rescaled hopping amplitudes, we are able to employ an orthogonal basis transformation to exactly map it on a set of decoupled linear chains with sizes and degeneracies written in terms of the network branching parameter q and generation number n. The integrated density of states is shown to have a fractal structure of gaps and degeneracies with a power-law decay at the band bottom. The spectral dimension d(s) coincides with the network topological dimension d(f) = ln(2q)/ln(2). We perform a finite-size scaling analysis of the fraction of condensed particles and specific heat to characterize the critical behavior of the BEC transition that occurs for q > 2 (d(s) > 2). The critical exponents are shown to follow those for lattices with a pure power-law spectral density, with non-mean-field values for q < 8 (d(s) < 4). The transition temperature is shown to grow monotonically with the branching parameter, obeying the relation 1/T(c) = a + b/(q - 2).