Linearized spectral decimation in fractals

Iliasov, Askar A.; Katsnelson, M. I.; Yuan, Shengjun

doi:10.1103/physrevb.102.075440

Cited by 5 publications

(4 citation statements)

References 39 publications

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“…We also note that the renormalization-group approach has been used in 1D Fibonacci chains, 2D lattices such as Vicsek [8] and other fractals [56,96], and Penrose tiling [97]. However, the decimating procedures in the SC(n, m, g * ) lattices become very tedious due to their huge lattice size.…”

Section: Discussionmentioning

confidence: 99%

“…We now discuss the other tool, the level-spacing distribution (LSD) between the adjacent energy levels. As a prelude to this work, Iliasov et al [55,56] studied the LSD in two simplified iterated structures (each square-and triangle-block unit with fewer connections [56], respectively). They exploited the decimating procedure, and gave the analytical asymptotic formula P(s) ∼ s α at the decay tail, where s is the nearest-neighbor level spacing, α is a constant, and P(s) is the distribution.…”

Section: Introductionmentioning

confidence: 99%

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Energy-level statistics in planar fractal tight-binding models

Yao

Yang

Iliasov³

et al. 2023

Phys. Rev. B

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In this study, we examine the statistics of level spectra in a noninteracting electron gas confined to a Sierpiński carpet lattice. These lattices are constructed using two types of the self and gene patterns, and they are categorized by the area-perimeter scaling law. The singularly continuous spectra, along with the nearest level spacing distribution and gap-ratio distribution, reveal a critical phase different from both extended and localized phases. This critical phase differs from the behavior near the Anderson model's metal-insulator transition. The Wigner-like conjecture is confirmed for both lattice classes, indicating Gaussian orthogonal symmetry. A similar observation was made in a quasiperiodic lattice [Phys. Rev. Lett. 80, 3996 (1998)]. The self-similar nature of fractals also leads to level clustering behavior.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Energy-level statistics in planar fractal tight-binding models

Yao

Yang

Iliasov³

et al. 2023

Phys. Rev. B

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“…A fractal has a hierarchically self-similar block structure quantified by the noninteger Hausdorff dimension d H [1][2][3][4]. The unique self-similarity endows fractal nanostructures with a wealth of exotic and interesting physical features on electronic energy spectrum statistics [5][6][7][8], quantum transport properties [9][10][11][12][13][14][15][16], plasmons [17], flat bands [18][19][20][21], and topological phases [22][23][24][25][26]. Recently, nanoscale fractal structures, such as Sierpinski carpets (SC) and gaskets with atoms or molecules as building units, have been achieved by the bottom-up nanofabrication methods, including molecular selfassembly [27][28][29][30][31][32][33], chemical reactions [34], template packings [35], and atomic manipulations in a scanning tunneling microscope [36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Electronic properties and quantum transport in functionalized graphene Sierpinski-carpet fractals

et al. 2022

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Recent progress in the controllable functionalization of graphene surfaces enables the experimental realization of complex functionalized graphene nanostructures, such as Sierpinski carpet (SC) fractals. Herein, we model the SC fractals formed by hydrogen and fluorine functionalized patterns on graphene surfaces, namely, H-SC and F-SC, respectively. We then reveal their electronic properties and quantum transport features. From the calculated results of the total and local densities of state, we find that states in H-SC and F-SC have two characteristics: (i) low-energy states inside about |E /t| 1 (with t as the nearest-neighbor hopping) are localized inside free graphene regions due to the insulating properties of functionalized graphene regions and (ii) high-energy states in F-SC have two special energy ranges including −2.3 < E /t < −1.9 with localized holes only inside free graphene areas and 3 < E /t < 3.7 with localized electrons only inside fluorinated graphene areas. The two characteristics are further verified by the real-space distributions of normalized probability density. We analyze the fractal dimension of their quantum conductance spectra and find that conductance fluctuations in these structures follow the Hausdorff dimension. We calculate their optical conductivity and find that several additional conductivity peaks appear in high-energy ranges due to the adsorbed H or F atoms.

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“…A fractal, with non-integer Hausdorff dimension d H [1][2][3][4], has a hierarchically self-similar structure. The intrinsic features from this, including electronic energy spectrum statistics and transport [5][6][7][8][9][10][11][12][13][14], quantum Hall effect [15,16], plasmon [17], flat bands [18][19][20][21], and topological properties [22][23][24][25][26] have attracted dense interest. For example, d H determines the box-counting dimension of quantum conductance fluctuations in Sierpinski carpet geometry with infinite ramification number [2,10], and there are sharp peaks in the optical spectrum due to the specific electronic state pairs [11].…”

Section: Introductionmentioning

confidence: 99%

Electronic properties and quantum transports in functionalized graphene Sierpinski carpet fractals

Yang,

Zhou,

Yao

et al. 2021

Preprint

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We propose a way to create effective plane fractals for confined electrons by chemical surface doping in two-dimensional materials. In particular, for Sierpinski carpet (SC), we consider the adsorption of hydrogen or fluorine atoms on monolayer graphene, forming an effective fractional geometry (H-SC and F-SC). By calculating the spatial distribution of states and the fractal analysis of quantum conductance fluctuations of H-SC, F-SC, and real SC, we find there is a consistency of the electronic states in these different systems. Further optical calculations verified the fractional confinement of electrons in H-SC and F-SC. Our results indicate that the chemical surface doping on graphene is an efficient and accessible approach to confine the electrons roaming in a fractional dimension, and will stimulate further theoretical and experimental studies of fractals based on twodimensional materials.

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Linearized spectral decimation in fractals

Cited by 5 publications

References 39 publications

Energy-level statistics in planar fractal tight-binding models

Energy-level statistics in planar fractal tight-binding models

Electronic properties and quantum transport in functionalized graphene Sierpinski-carpet fractals

Electronic properties and quantum transports in functionalized graphene Sierpinski carpet fractals

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