Bose-Einstein condensation in the Apollonian complex network

Random Apollonian networks with tailored clustering coefficient

We introduce a family of complex networks that interpolates between the Apollonian network and its binary version, recently introduced in E. M. K. Souza et al. [Phys. Rev. E 107, 024305 (2023)2470-004510.1103/PhysRevE.107.024305], via random removal of nodes. The dilution process allows the clustering coefficient to vary from C=0.828 to C=0 while maintaining the behavior of average path length and other relevant quantities as in the deterministic Apollonian network. Robustness against the random deletion of nodes is also reported on spectral quantities such as the ground-state localization degree and its energy gap to the first excited state. The loss of the 2π/3 rotation symmetry as a treelike network emerges is investigated in the light of the hub wavefunction amplitude. Our findings expose the interplay between the small-world property and other distinctive traits exhibited by Apollonian networks, as well as their resilience against random attacks.

Binary Apollonian networks

There is a well-known relationship between the binary Pascal's triangle and the Sierpinski triangle, in which the latter is obtained from the former by successive modulo 2 additions starting from a corner. Inspired by that, we define a binary Apollonian network and obtain two structures featuring a kind of dendritic growth. They are found to inherit the small-world and scale-free properties from the original network but display no clustering. Other key network properties are explored as well. Our results reveal that the structure contained in the Apollonian network may be employed to model an even wider class of real-world systems.

Vertex Labeling and Routing for Farey-Type Symmetrically-Structured Graphs

The generalization of Farey graphs and extended Farey graphs all originate from Farey graphs. They are simultaneously scale-free and small-world. A labeling of the vertices for them are proposed here. All of the shortest paths between any two vertices in these two graphs can be determined only on their labels. The number of shortest paths between any two vertices is the product of two Fibonacci numbers; it is increasing almost linearly with the order or size of the graphs. However, the label-based routing algorithm runs in logarithmic time O(logn). Our efficient routing protocol for Farey-type models should help contribute toward the understanding of several physical dynamic processes.

Ferromagnetic model on the Apollonian packing

This work investigates the influence of geometrical features of the Apollonian packing (AP) on the behavior of magnetic models. The proposed model differs from previous investigations on the Apollonian network (AN), where the magnetic coupling constants depend only on the properties of the network structure defined by the packing, but not on quantitative aspects of its geometry. In opposition to the exact scale invariance observed in the AN, the circle's sizes in the AP are scaled by different factors when one goes from one generation to the next, requiring a different approach for the evaluation of the model's properties. If the nearest-neighbors coupling constants are defined by J_{i,j}∼1/(r_{i}+r_{j})^{α}, where r_{i} indicates the radius of the circle i containing the node i, the results for the correlation length ξ indicate that the model's behavior depend on α. In the thermodynamic limit, the uniform model (α=0) is characterized by ξ→∞ for all T>0. Our results indicate that, on increasing α, the system changes to an uncorrelated pattern, with finite ξ at all T>0, at a value α_{c}≃0.743. For any fixed value of α, no finite temperature singularity in the specific heat is observed, indicating that changes in the magnetic ordering occur only when α is changed. This is corroborated by the results for the magnetization and magnetic susceptibility.

The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks

To depict the complex relationship among nodes and the evolving process of a complex system, a Bose-Einstein hypernetwork is proposed in this paper. Based on two basic evolutionary mechanisms, growth and preference jumping, the distribution of hyperedge cardinalities is studied. The Poisson process theory is used to describe the arrival process of new node batches. And, by using the Poisson process theory and a continuity technique, the hypernetwork is analyzed and the characteristic equation of hyperedge cardinalities is obtained. Additionally, an analytical expression for the stationary average hyperedge cardinality distribution is derived by employing the characteristic equation, from which Bose-Einstein condensation in the hypernetwork is obtained. The theoretical analyses in this paper agree with the conducted numerical simulations. This is the first study on the hyperedge cardinality in hypernetworks, where Bose-Einstein condensation can be regarded as a special case of hypernetworks. Moreover, a condensation degree is also discussed with which Bose-Einstein condensation can be classified.

Complex networks from space-filling bearings

Two-dimensional space-filling bearings are dense packings of disks that can rotate without slip. We consider the entire first family of bearings for loops of four disks and propose a hierarchical construction of their contact network. We provide analytic expressions for the clustering coefficient and degree distribution, revealing bipartite scale-free behavior with a tunable degree exponent depending on the bearing parameters. We also analyze their average shortest path and percolation properties.

Soliton transport in tubular networks: transmission at vertices in the shrinking limit

Soliton transport in tubelike networks is studied by solving the nonlinear Schrödinger equation (NLSE) on finite thickness ("fat") graphs. The dependence of the solution and of the reflection at vertices on the graph thickness and on the angle between its bonds is studied and related to a special case considered in our previous work, in the limit when the thickness of the graph goes to zero. It is found that both the wave function and reflection coefficient reproduce the regime of reflectionless vertex transmission studied in our previous work.

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).

Ising models on the regularized Apollonian network

We investigate the critical properties of Ising models on a regularized Apollonian network (RAN), here defined as a kind of Apollonian network in which the connectivity asymmetry associated with its corners is removed. Different choices for the coupling constants between nearest neighbors are considered and two different order parameters are used to detect the critical behavior. While ordinary ferromagnetic and antiferromagnetic models on a RAN do not undergo a phase transition, some antiferrimagnetic models show an interesting infinite-order transition. All results are obtained by an exact analytical approach based on iterative partial tracing of the Boltzmann factor as an intermediate step for the calculation of the partition function and the order parameters.

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 d(s)=2ln(3)/ln(9/5)~/=3.74.

Phase transition of light on complex quantum networks

Recent advances in quantum optics and atomic physics allow for an unprecedented level of control over light-matter interactions, which can be exploited to investigate new physical phenomena. In this work we are interested in the role played by the topology of quantum networks describing coupled optical cavities and local atomic degrees of freedom. In particular, using a mean-field approximation, we study the phase diagram of the Jaynes-Cummings-Hubbard model on complex networks topologies, and we characterize the transition between a Mott-like phase of localized polaritons and a superfluid phase. We found that, for complex topologies, the phase diagram is nontrivial and well defined in the thermodynamic limit only if the hopping coefficient scales like the inverse of the maximal eigenvalue of the adjacency matrix of the network. Furthermore we provide numerical evidences that, for some complex network topologies, this scaling implies an asymptotically vanishing hopping coefficient in the limit of large network sizes. The latter result suggests the interesting possibility of observing quantum phase transitions of light on complex quantum networks even with very small couplings between the optical cavities.

Geometrical and Anderson transitions in harmonic chains with constrained long-range couplings

Low-dimensional systems with long-range couplings usually present phase transitions which are absent in the short-ranged counterpart model. In this work, we show that a harmonic chain with long-range couplings restricted by a cost function proportional to the chain length N exhibits two distinct phase transitions. In the present model, two sites at a distance r>1 are connected by a spring with probability 1/r(α) with the constraint that the total length of the non-nearest-neighbor couplings is limited to λN, where λ is a cost parameter. A geometrical phase transition is found at α=1.5 between a phase with a finite number of long-range couplings and a phase on which the number of long-range couplings is proportional to the system size. Further, the normal vibrational modes of this chain display a phase transition from delocalized to localized modes at a smaller value of α. Maximum effective disorder is reached at α=2 for which the frequency of the lowest vibrational mode exhibits a pronounced peak.

Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation

We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one [Sobirov et al., Phys. Rev. E 81, 066602 (2010)] on the continuum NLSE to the discrete case. We find (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

Bose-Einstein condensation in the infinitely ramified star and wheel graphs

In this work, we provide exact solutions for the ideal boson lattice gas on the infinitely ramified star and wheel graphs. Within a tight-binding description, we show that Bose-Einstein condensation (BEC) takes place at a finite temperature after a proper rescaling of the hoping integral ɛ connecting a central site to the peripheral ones. Analytical expressions for the transition temperature, the condensed gas fraction, and the specific heat are given for the star graph as a function of the density of particles n. In particular, the specific heat has a mean-field character, being null in the high-temperature noncondensed phase with a discontinuity at T(c). In the wheel graph, on which the peripheral sites form a closed chain with hopping integral t, BEC takes place only above a critical value of the ratio ɛ/t for which a gap ΔE appears between the ground state and a one-dimensional band. A detailed analysis of the BEC characteristics as a function of n and ΔE is provided. The specific heat in the high-temperature phase of the wheel graph remains finite due to correlations among the peripheral sites.

q-state Potts model on the Apollonian network

The q-state Potts model is studied on the Apollonian network with Monte Carlo simulations and the transfer matrix method. The spontaneous magnetization, correlation length, entropy, and specific heat are analyzed as a function of temperature for different number of states, q. Different scaling functions in temperature and q are proposed. A quantitative agreement is found between results from both methods. No critical behavior is observed in the thermodynamic limit for any number of states.