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Fractal-driven universality in quantum transport through mesoscopic devices

F. M. Oliveira1,2, Louis G. C. S. Sá1, and J. G. G. S. Ramos1

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Phys. Rev. E 113, 064131 – Published 15 June, 2026

DOI: https://doi.org/10.1103/h6yp-4fty

Abstract

We examine the impact of spatial fractality on universal quantum transport in mesoscopic systems featuring Sierpiński carpet scattering geometries. Transport calculations are performed using the nonequilibrium Green's function formalism within the Landauer-Büttiker framework, allowing for a detailed statistical characterization of conductance and shot noise power fluctuations. Our analysis reveals that the fractal dimension of these fluctuations increases systematically with the iteration depth of the structure, reflecting enhanced complexity in quantum interference patterns. A direct and robust correlation is established between the Hausdorff dimension of the scattering region and the effective fractal dimension of transport observables, independent of the Wigner-Dyson universal symmetry class (GOE, GUE, and GSE). This provides compelling evidence that scattering across internal lacunae constitutes the dominant mechanism shaping transport behavior in the universal regime. Moreover, we find that the density of local extrema increases while the correlation length decreases with increasing fractality; however, the number of extrema per correlation length remains remarkably with negligible variation. Notably, the autocorrelation functions progressively deviate from the standard Lorentzian form, indicating that geometric self-similarity introduces nontrivial modifications to the spectral correlations predicted by Random Matrix Theory.

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References (61)

  1. F. Haake, Quantum Signatures of Chaos, 3rd ed. (Springer, Berlin, 2010).
  2. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
  3. T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Random-matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  4. N. Bohr, Neutron capture and nuclear constitution, Nature (London) 137, 344 (1936).
  5. C. Mahaux and H. A. Weidenmüller, Shell-Model Approach to Nuclear Reactions (North-Holland, Amsterdam, 1969).
  6. E. P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Proc. Cambridge Philos. Soc. 47, 790 (1951).
  7. G. E. Mitchell, A. Richter, and H. A. Weidenmüller, Random matrices and chaos in nuclear physics: Nuclear reactions, Rev. Mod. Phys. 82, 2845 (2010).
  8. A. M. M. Argüello, M. Carrera-Núñez, and J. A. Méndez-Bermúdez, Scattering and transport properties of the three classical Wigner–Dyson ensembles at the Anderson transition, Phys. Rev. E 107, 024139 (2023).
  9. A. K. Das and A. Ghosh, Nonergodic extended states in the β ensemble, Phys. Rev. E 105, 054121 (2022).
  10. G. Cipolloni, L. Erdős, and D. Schröder, On the spectral form factor for random matrices, Commun. Math. Phys. 401, 1665 (2023).
  11. M. Hopjan and L. Vidmar, Scale-invariant critical dynamics at eigenstate transitions, Phys. Rev. Lett. 131, 060404 (2023).
  12. C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69, 731 (1997).
  13. Y. Alhassid, The statistical theory of quantum dots, Rev. Mod. Phys. 72, 895 (2000).
  14. T. Heinzel, Mesoscopic Electronics in Solid State Nanostructures (Wiley-VCH, Weinheim, 2003).
  15. K. Kawabata, Z. Xiao, T. Ohtsuki, and R. Shindou, Singular-value statistics of non-Hermitian random matrices and open quantum systems, PRX Quantum 4, 040312 (2023).
  16. A. M. García-García, L. Sá, and J. J. M. Verbaarschot, Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev–Ye–Kitaev model, Phys. Rev. D 107, 066007 (2023).
  17. F. Roccati, F. Balducci, R. Shir, and A. Chenu, Diagnosing non-Hermitian many-body localization and quantum chaos via singular value decomposition, Phys. Rev. B 109, L140201 (2024).
  18. R. Duda, M. N. Ivaki, I. Sahlberg, K. Pöyhönen, and T. Ojanen, Quantum walks on random lattices: Diffusion, localization, and the absence of parametric quantum speedup, Phys. Rev. Res. 5, 023150 (2023).
  19. P. A. Lee, A. D. Stone, and H. Fukuyama, Universal conductance fluctuations in metals: Effects of finite temperature, interactions, and magnetic field, Phys. Rev. B 35, 1039 (1987).
  20. J. G. G. S. Ramos, D. Bazeia, M. S. Hussein, and C. H. Lewenkopf, Conductance peaks in open quantum dots, Phys. Rev. Lett. 107, 176807 (2011).
  21. L. G. C. S. Sá, A. L. R. Barbosa, and J. G. G. S. Ramos, Conductance peak density in disordered graphene topological insulators, Phys. Rev. B 102, 115105 (2020).
  22. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier, Amsterdam, 2004).
  23. R. Landauer, Spatial variation of currents and fields due to localized scatterers in metallic conduction, IBM J. Res. Dev. 1, 223 (1957).
  24. M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett. 57, 1761 (1986).
  25. R. Blümel and U. Smilansky, Classical irregular scattering and its quantum-mechanical implications, Phys. Rev. Lett. 60, 477 (1988).
  26. R. A. Jalabert, H. U. Baranger, and A. D. Stone, Conductance fluctuations in the ballistic regime: A probe of quantum chaos? Phys. Rev. Lett. 65, 2442 (1990).
  27. K. B. Efetov, Temperature effects in quantum dots in the regime of chaotic dynamics, Phys. Rev. Lett. 74, 2299 (1995).
  28. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1982).
  29. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd ed. (John Wiley & Sons, Chichester, 2014).
  30. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed. (Westview Press, Boulder, 2015).
  31. S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh, I. Swart, and C. M. Smith, Design and characterization of electrons in a fractal geometry, Nat. Phys. 15, 127 (2019).
  32. L. Lage and A. Latgé, Electronic fractal patterns in building Sierpinski-triangles molecular systems, Phys. Chem. Chem. Phys. 24, 19576 (2022).
  33. B. Pal and K. Saha, Flat bands in fractal-like geometry, Phys. Rev. B 97, 195101 (2018).
  34. D. Leykam, A. Andreanov, and S. Flach, Artificial flat band systems: From lattice models to experiments, Adv. Phys. X. 3, 1473052 (2018).
  35. X. Zhang, G. Gu, N. Li, H. Wang, H. Tang, Y. Zhang, S. Hou, and Y. Wang, One-dimensional molecular chains formed by Sierpiński triangles on Au(111), RSC Adv. 8, 1852 (2018).
  36. F. L. Sendker, Y. K. Lo, T. Heimerl, S. Bohn, L. J. Persson, C.-N. Mais, W. Sadowska, N. Paczia, E. Nußbaum, M. C. Sánchez Olmos, K. Forchhammer, D. Schindler, T. J. Erb, J. L. P. Benesch, E. G. Marklund, G. Bange, J. M. Schuller, and G. K. A. Hochberg, Emergence of fractal geometries in the evolution of a metabolic enzyme, Nature (London) 628, 894 (2024).
  37. A. Gowrisankar and S. Banerjee, Frontiers of fractals for complex systems: Recent advances and future challenges, Eur. Phys. J.: Spec. Top. 230, 3743 (2021).
  38. J. Dai, X. Zhao, Z. Peng, J. Li, Y. Lin, X. Wen, L. Xing, W. Zhao, J. Shang, Y. Wang, J. Liu, and K. Wu, Assembling surface molecular Sierpiński triangle fractals via K+-invoked electrostatic interaction, J. Am. Chem. Soc. 145, 13531, (2023).
  39. H. Hennig, R. Fleischmann, L. Hufnagel, and T. Geisel, Fractal conductance fluctuations of classical origin, Phys. Rev. E 76, 015202(R) (2007).
  40. X.-Y. Xu, X.-W. Wang, D.-Y. Chen, C. M. Smith, and X.-M. Jin, Quantum transport in fractal networks, Nat. Photon. 15, 703 (2021).
  41. E. van Veen, S. Yuan, M. I. Katsnelson, M. Polini, and A. Tomadin, Quantum transport in Sierpinski carpets, Phys. Rev. B 93, 115428 (2016).
  42. A. K. Slimane, P. Reck, and G. Fleury, Simulating time-dependent thermoelectric transport in quantum systems, Phys. Rev. B 101, 235413 (2020).
  43. M. M. Odashima and C. H. Lewenkopf, Time-dependent resonant tunneling transport: Keldysh and Kadanoff–Baym nonequilibrium Green's functions in an analytically soluble problem, Phys. Rev. B 95, 104301 (2017).
  44. Z. Yu, L. Zhang, Y. Xing, and J. Wang, Investigation of transient heat current from first principles using complex absorbing potential, Phys. Rev. B 90, 115428 (2014).
  45. M. Ridley, M. Galperin, E. Gull, and G. Cohen, Numerically exact full counting statistics of the energy current in the Kondo regime, Phys. Rev. B 100, 165127 (2019).
  46. Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University Press, Cambridge, 2009).
  47. C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal, Kwant: A software package for quantum transport, New J. Phys. 16, 063065 (2014).
  48. Ya M. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Rep. 336, 1 (2000).
  49. M. Büttiker, Scattering theory of current and intensity noise correlations in conductors and wave guides, Phys. Rev. B 46, 12485 (1992).
  50. Th Martin and R. Landauer, Wave-packet approach to noise in multichannel mesoscopic systems, Phys. Rev. B 45, 1742 (1992).
  51. Y. V. Nazarov and M. Kindermann, Full counting statistics of a general quantum mechanical variable, Eur. Phys. J. B 35, 413 (2003).
  52. M. Büttiker, Quantized transmission of a saddle-point constriction, Phys. Rev. B 41, 7906 (1990).
  53. E. van Veen, A. Tomadin, M. Polini, M. I. Katsnelson, and S. Yuan, Optical conductivity of a quantum electron gas in a Sierpinski carpet, Phys. Rev. B 96, 235438 (2017).
  54. X. Yang, W. Zhou, Q. Yao, P. Lv, Y. Wang, and S. Yuan, Electronic properties and quantum transport in functionalized graphene Sierpinski-carpet fractals, Phys. Rev. B 105, 205433 (2022).
  55. Y. Wang, W. Zhou, Q. Yao, P. Lv, and S. Yuan, Electronic states and quantum transport in bilayer graphene Sierpinski-carpet fractals, Phys. Rev. B 109, 235429 (2024).
  56. P. A. Lee and A. D. Stone, Universal conductance fluctuations in metals, Phys. Rev. Lett. 55, 1622 (1985).
  57. T. Kiss and I. Jex, Photons walk on fractal graphs, Nat. Photon. 15, 641 (2021).
  58. A. Rojo-Francàs, P. Pansari, U. Bhattacharya, B. Juliá-Díaz, and T. Grass, Anomalous quantum transport in fractal lattices, Commun. Phys. 7, 259 (2024).
  59. Q. Yao, X. Yang, A. A. Iliasov, M. I. Katsnelson, and S. Yuan, Energy-level statistics in planar fractal tight-binding models, Phys. Rev. B 107, 115424 (2023).
  60. Q. Yao, X. Yang, A. A. Iliasov, M. I. Katsnelson, and S. Yuan, Wave functions in the critical phase: A planar Sierpiński fractal lattice, Phys. Rev. B 110, 035403 (2024).
  61. D. Bazeia, M. B. P. N. Pereira, A. V. Brito, B. F. de Oliveira, and J. G. G. S. Ramos, A novel procedure for the identification of chaos in complex biological systems, Sci. Rep. 7, 44900 (2017).

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Phys. Rev. E 113, 064131– Published 15 June, 2026
  • Received 12 July 2025
  • Revised 10 March 2026
  • Accepted 28 May 2026
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