Bose-Einstein condensation in diamond hierarchical lattices

Abstract

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).

Josephson Currents and Gap Enhancement in Graph Arrays of Superconductive Islands

“…Theoretical analyses have reported on the interest of the structures that we have investigated from the fundamental mathematic point of view [ 14 , 27 , 28 , 29 ], and other publications [ 22 , 30 , 31 , 32 ] have demonstrated the interest from different perspectives for the type of systems we have investigated. It is worth pointing out that the phenomena that we observe are not just determining an increase of currents along specific directions of the graphs when the superconductors are in the condensed state: our results point toward the existence of new topological states in which fundamental properties of the superconducting state, such as gap and transition temperature, can be conditioned.…”

Conditioning of Superconductive Properties in Graph-Shaped Reticles

“…The predictions of the theory are concerned with thermal hopping of bosons (Cooper pairs), via Josephson tunnelling, between superconductive islands which can take place when the thermal excitations energy is comparable with Josephson potentials all over the graph array distribution 1,8 . Considered that experiments on samples fabricated in conventional superconductive technologies are typically performed at about 4.2 K and below, this condition imposes limitations on the amplitude of the Josephson currents I c and relative zero-bias energy E J = Φ 0 I c /2π 7,9 , where Φ 0 = 2.07 × 10 −15 Wb is the flux quantum. Therefore the estimate of the currents, for the zero-bias case, is given by I c = (2π/Φ 0 )k B T. Substituting numerical values, with k B = 1.38 × 10 −23 J/K (Boltzmann's constant) and T = 4.2 K, in the last equation we get I c = 176 nA.…”

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

“…Exact calculations on hierarchical lattices are also currently widely used in a variety of statistical mechanics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], finance [37], and, most recently, DNA-binding [38] problems.…”