- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Harris-Luck criterion in the plateau transition of the integer quantum Hall effect

H. Topchyan1, W. Nuding2, A. Klümper2, and A. Sedrakyan1

PDF

Phys. Rev. B 111, L100201 – Published 12 March, 2025

DOI: https://doi.org/10.1103/PhysRevB.111.L100201

Abstract

The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension d and localization length exponent ν. It states that the new disorder can be relevant only if dν<2. We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Klümper, W. Nuding, and A. Sedrakyan, Phys. Rev. B 95, 125414 (2017)] and show that the fluctuations of the geometry are relevant despite dν>2, implying that the Harris criterion should be modified. We have observed that the fluctuations of the critical point in different quenched configurations of disordered network blocks is of order L0, i.e., it does not depend on block size L in contrast to the expectation based on the Harris criterion that they should decrease as L−d/2 according to the central limit theorem. Since L0>(x−xc) is always satisfied near the critical point, the mentioned network disorder is relevant and the critical indices of the system can be changed. We have also shown that the GKNS disordered network is fundamentally different from Voronoi-Delaunay and dynamically triangulated random lattices: The probability of higher connectivity in the GKNS network decreases in a power law as opposed to an exponential, indicating that we are dealing with a “scale-free” network, such as the internet, protein-protein interactions, etc.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C: Solid State Phys. 7, 1671 (1974).
  2. J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer, Finite-size scaling and correlation lengths for disordered systems, Phys. Rev. Lett. 57, 2999 (1986).
  3. I. A. Gruzberg, A. Klümper, W. Nuding, and A. Sedrakyan, Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions, Phys. Rev. B 95, 125414 (2017).
  4. A. Klümper, W. Nuding, and A. Sedrakyan, Random network models with variable disorder of geometry, Phys. Rev. B 100, 140201(R) (2019).
  5. H. Topchyan, I. A. Gruzberg, W. Nuding, A. Klümper, and A. Sedrakyan, Integer quantum Hall transition: An S-matrix approach to random networks, Phys. Rev. B 110, L081112, (2024).
  6. J. T. Chalker and P. D. Coddington, Percolation, quantum tunnelling and the integer Hall effect, J. Phys. C: Solid State Phys. 21, 2665 (1988).
  7. K. Slevin and T. Ohtsuki, Critical exponent for the quantum Hall transition, Phys. Rev. B 80, 041304(R) (2009).
  8. M. Amado, A. V. Malyshev, A. Sedrakyan, and F. Domínguez-Adame, Numerical study of the localization length critical index in a network model of plateau-plateau transitions in the quantum Hall effect, Phys. Rev. Lett. 107, 066402 (2011).
  9. J. P. Dahlhaus, J. M. Edge, J. Tworzydło, and C. W. J. Beenakker, Quantum Hall effect in a one-dimensional dynamical system, Phys. Rev. B 84, 115133 (2011).
  10. I. C. Fulga, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Topological quantum number and critical exponent from conductance fluctuations at the quantum Hall plateau transition, Phys. Rev. B 84, 245447 (2011).
  11. H. Obuse, I. A. Gruzberg, and F. Evers, Finite-size effects and irrelevant corrections to scaling near the integer quantum Hall transition, Phys. Rev. Lett. 109, 206804 (2012).
  12. K. Slevin and T. Ohtsuki, Finite size scaling of the Chalker-Coddington model, Int. J. Mod. Phys. Conf. Ser. 11, 60 (2012).
  13. W. Nuding, A. Klümper, and A. Sedrakyan, Localization length index and subleading corrections in a Chalker-Coddington model: A numerical study, Phys. Rev. B 91, 115107 (2015).
  14. A. Weinrib and B. I. Halperin, Critical phenomena in systems with long-range-correlated quenched disorder, Phys. Rev. B 27, 413 (1983).
  15. V. V. Prudnikov and A. A. Fedorenko, Critical behaviour of 3D systems with long-range correlated quenched defects, J. Phys. A: Math. Gen. 32, L399 (1999).
  16. J. M. Luck, A classification of critical phenomena on quasi-crystals and other aperiodic structures, Europhys. Lett. 24, 359 (1993).
  17. A. Okabe, B. N. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations, 2nd ed., Wiley Series in Probability and Statistics (Wiley, Chichester, UK, 2000).
  18. G. Schliecker, Structure and dynamics of cellular systems, Adv. Phys. 51, 1319 (2002).
  19. F. Pázmándi, R. T. Scalettar, and G. T. Zimányi, Revisiting the theory of finite size scaling in disordered systems: ν can be less than 2/d, Phys. Rev. Lett. 79, 5130 (1997).
  20. A. Aharony, A. B. Harris, and S. Wiseman, Critical disordered systems with constraints and the inequality ν>2/d, Phys. Rev. Lett. 81, 252 (1998).
  21. A. L. Korzhenevskii, H.-O. Heuer, and K. Herrmanns, Percolative phase transition in a disordered Ising model with finite disorder correlation length, J. Phys. A: Math. Gen. 31, 927 (1998).
  22. J. Ambjørn, B. Durhuus, and T. Jonsson, Quantum Geometry: A Statistical Field Theory Approach (Cambridge University Press, Cambridge, UK, 1997).
  23. D. Espriu, M. Gross, P. Rakow, and J. Wheater, Continuum limit on a 2-dimensional random lattice, Nucl. Phys. B 265, 92 (1986).
  24. W. Janke, M. Katoot, and R. Villanova, Ising model universality for two-dimensional lattices, Phys. Lett. B 315, 412 (1993).
  25. W. Janke, M. Katoot, and R. Villanova, Single-cluster Monte Carlo study of the Ising model on two-dimensional random lattices, Phys. Rev. B 49, 9644 (1994).
  26. W. Janke and R. Villanova, Two-dimensional eight-state Potts model on random lattices: A Monte Carlo study, Phys. Lett. A 209, 179 (1995).
  27. K. N. Anagnostopoulos, P. Bialas, and G. Thorleifsson, The Ising model on a quenched ensemble of c=−5 gravity graphs, J. Stat. Phys. 94, 321 (1999).
  28. W. Janke and D. A. Johnston, Ising and Potts models on quenched random gravity graphs, Nucl. Phys. B 578, 681 (2000).
  29. W. Janke and D. A. Johnston, Non-self-averaging in autocorrelations for Potts models on quenched random gravity graphs, J. Phys. A: Math. Gen. 33, 2653 (2000).
  30. W. Janke and R. Villanova, Ising model on three-dimensional random lattices: A Monte Carlo study, Phys. Rev. B 66, 134208 (2002).
  31. W. Janke and M. Weigel, Harris-Luck criterion for random lattices, Phys. Rev. B 69, 144208 (2004).
  32. H. Barghathi and T. Vojta, Phase transitions on random lattices: How random is topological disorder? Phys. Rev. Lett. 113, 120602 (2014).
  33. M. M. de Oliveira and S. C. Ferreira, Universality of the contact process with random dilution, J. Stat. Mech. (2008) P11001.
  34. M. Schrauth, J. S. E. Portela, and F. Goth, Violation of the Harris-Barghathi-Vojta criterion, Phys. Rev. Lett. 121, 100601 (2018).
  35. R. Conti, H. Topchyan, R. Tateo, and A. Sedrakyan, Geometry of random potentials: Induction of two-dimensional gravity in quantum Hall plateau transitions, Phys. Rev. B 103, L041302 (2021).
  36. J. Drouffe and C. Itzykson, Random geometry and the statistics of two-dimensional cells, Nucl. Phys. B 235, 45 (1984).
  37. R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
  38. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286, 509 (1999).
  39. R. Albert, H. Jeong, and A.-L. Barabási, Diameter of the World-Wide Web, Nature (London) 401, 130 (1999).
  40. M. Faloutsos, P. Faloutsos, and C. Faloutsos, On power-law relationships of the internet topology, SIGCOMM Comput. Commun. Rev. 29, 251 (1999).
  41. H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A.-L. Barabási, The large-scale organization of metabolic networks, Nature (London) 407, 651 (2000).
  42. D. Boulatov, V. Kazakov, I. Kostov, and A. Migdal, Analytical and numerical study of a model of dynamically triangulated random surfaces, Nucl. Phys. B 275, 641 (1986).
  43. V. Kazakov, Exactly solvable Potts models, bond- and tree-like percolation on dynamical (random) planar lattice, Nucl. Phys. B 4, 93 (1988).
  44. V. Kazakov and A. Migdal, Recent progress in the theory of noncritical strings, Nucl. Phys. B 311, 171 (1988).
  45. V. Kazakov, Percolation on a fractal with the statistics of planar Feynman graphs: Exact solution, Mod. Phys. Lett. A 04, 1691 (1989).
  46. V. Knizhnik, A. Polyakov, and A. Zamolodchikov, Fractal structure of 2d–quantum gravity, Mod. Phys. Lett. A 03, 819 (1988).
  47. S. Bera, J. Dieplinger, and N. P. Nayak, Quantum Hall criticality in an amorphous Chern insulator, Phys. Rev. B 109, 174213 (2024).

Outline

Information

Phys. Rev. B 111, L100201– Published 12 March, 2025
  • Received 11 November 2024
  • Revised 4 February 2025
  • Accepted 5 March 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation