Free-electron gas in the Apollonian network: Multifractal energy spectrum and its thermodynamic fingerprints

Oliveira, I. N. de; Moura, F. A. B. F. de; Lyra, M. L.; Andrade, José S.; Albuquerque, E.L.

doi:10.1103/physreve.79.016104

Cited by 42 publications

(31 citation statements)

References 54 publications

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“…In fact, this type of techniques has been considered for different hierarchical structures to study critical phenomena [10][11][12][13][14]. This important property has also motivated using AN, as a first approximation, to study problems from several different areas, such as physiology, geology, communication, energy, and fluid transport [15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

q-state Potts model on the Apollonian network

2010

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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.

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Section: Introductionmentioning

confidence: 99%

q-state Potts model on the Apollonian network

2010

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“…Moreover, attention has been addressed to quantum processes on Apollonian networks [20,21], which provide an example of scale-free networks embedded in two dimensions. The quantum processes investigated are the Hubbard model [22], the free electron gas within the tight-binding model [23], and the topology induced BoseEinstein condensation in Apollonian networks [24]. The study of quantum phase transitions on these networks has attracted attention particularly in recent years motivated by the creation of a new self-similar macromolecule -a nanometer-scale Sierpinski hexagonal gasket [25].…”

mentioning

confidence: 99%

Phase diagram of the Bose-Hubbard model on complex networks

Halu¹,

Ferretti²,

Vezzani³

et al. 2012

EPL

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-Critical phenomena can show unusual phase diagrams when defined in complex network topologies. The case of classical phase transitions such as the classical Ising model and the percolation transition has been studied extensively in the last decade. Here we show that the phase diagram of the Bose-Hubbard model, an exclusively quantum mechanical phase transition, also changes significantly when defined on random scale-free networks. We present a mean-field calculation of the model in annealed networks and we show that when the second moment of the average degree diverges the Mott-insulator phase disappears in the thermodynamic limit. Moreover we study the model on quenched networks and we show that the Mott-insulator phase disappears in the thermodynamic limit as long as the maximal eigenvalue of the adjacency matrix diverges. Finally we study the phase diagram of the model on Apollonian scale-free networks that can be embedded in 2 dimensions showing the extension of the results also to this case.Introduction. -Recently great attention [1, 2] has been addressed to critical phenomena unfolding on complex networks. In this context it has been observed that the topology of the networks might significantly change the phase diagram of dynamical processes. For example when networks have a scale-free degree distribution P (k) ∼ k −λ and the second moment k 2 diverges with the network size, i.e. λ ∈ (2, 3] the Ising model [3][4][5][6], the percolation phase transition [7,8] and the epidemic spreading dynamics on annealed networks [9] are strongly affected. Moreover the spectral properties of the networks drive the epidemic spreading on quenched networks [10,11], the synchronization stability [12,13], the critical behavior of O(N ) models [14,15] and the critical fluctuations of an Ising model on spatial scale-free networks [16]. Quantum critical phenomena also might depend on the topology of the underlying lattice as it has been shown for Bose-Einstein condensation in heterogeneous networks [17]. Although large attention has been devoted to classical critical phenomena on scale-free networks, the behavior of quantum critical phenomena on scale-free networks has just started to be investigated.In particular the Anderson localization [18,19] was studied in complex networks showing that by modulating the clustering coefficient of the

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“…Despite the fact that several geometrical sets may present this property, we concentrate our investigation on the tight-binding model on the Apollonian network, because most of its quantum energy spectrum features, eigenstate localization properties, and quantum walk dynamics have been described in detail in several works [17,18,28,29].…”

Section: P K (E) For the Apollonian Networkmentioning

confidence: 99%

“…This network is obtained through the recursive application of a geometrical procedure, which leads to a series of network generations identified by a label g. Thus, the investigation of the tight-binding system requires the use of a sequence of Hamiltonian operators H g that account for all interactions between the available sites that have been introduced until generation g. Details of this kind of investigation can be found in Refs. [17,18], where an investigation of the properties of the eigenstates for successive generations of the Apollonian network has been carried out. The site labeling used herein has been introduced in Ref.…”

Section: The Modelmentioning

confidence: 99%

How the site degree influences quantum probability on inhomogeneous substrates

et al. 2017

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We investigate the effect of the node degree and energy E on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependence of the quantum probability for each site on its degree. For bi-regular structures, we prove analytically that the probability P k (E) of finding the particle on any site with k neighbors is independent of E. For more general structures, the dependency of P k (E) on E is discussed by taking into account exact results on a one-dimensional semi-regular chain: P k (E) is large for small values of E when k is also small, and its maximum values shift towards large values of |E| with increasing k. Numerical evaluations of P k (E) for two different types of percolation clusters and the Apollonian network suggest that this feature might be generally valid.

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Free-electron gas in the Apollonian network: Multifractal energy spectrum and its thermodynamic fingerprints

Cited by 42 publications

References 54 publications

q-state Potts model on the Apollonian network

q-state Potts model on the Apollonian network

Phase diagram of the Bose-Hubbard model on complex networks

How the site degree influences quantum probability on inhomogeneous substrates

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