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Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network

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  • Xie, Pinchen
  • Yang, Bingjia
  • Zhang, Zhongzhi
  • Andrade, Roberto F.S.

Abstract

A deterministic network with tree structure is considered, for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach. It amounts to successively increasing number of contributions at any finite step of construction of the tree, resulting in a causal chain. The resulting eigenvalues can be related the full energy spectrum of a nearest-neighbor tight-binding model defined on this structure. Given this association, it turns out that further properties of the eigenvectors can be evaluated, like the degree of quantum localization of the tight-binding eigenstates, expressed by the inverse participation ratio (IPR). It happens that, for the current model, the IPR’s are also suitable to be analytically expressed in terms in corresponding eigenvalue chain. The resulting IPR scaling behavior is expressed by the tails of eigenvalue chains as well.

Suggested Citation

  • Xie, Pinchen & Yang, Bingjia & Zhang, Zhongzhi & Andrade, Roberto F.S., 2018. "Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 40-48.
    • Handle: RePEc:eee:phsmap:v:502:y:2018:i:c:p:40-48
    DOI: 10.1016/j.physa.2018.02.089
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    References listed on IDEAS

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    1. Coronado, Ana V. & Carpena, Pedro, 2005. "Study of the log-periodic oscillations of the specific heat of Cantor energy spectra," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 358(2), pages 299-312.
    2. de Oliveira, I.N. & Lyra, M.L. & Albuquerque, E.L., 2004. "Specific heat anomalies of non-interacting fermions with multifractal energy spectra," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 424-432.
    3. Xie, Pinchen & Zhang, Zhongzhi & Comellas, Francesc, 2016. "On the spectrum of the normalized Laplacian of iterated triangulations of graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1123-1129.
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