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Class C quantum network model with random tunneling and its nonlinear sigma model representation

D. S. Katkov1,2, M. V. Parfenov2,3,4, and I. S. Burmistrov2,4

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Phys. Rev. B 114, 185405 – Published 2 September, 2026

DOI: https://doi.org/10.1103/5rk9-w686

Abstract

The spin quantum Hall effect is a relative of the integer quantum Hall effect, characterized by integer quantized spin Hall conductance. In this work, we formulate and investigate a quantum network model consisting of N channels per chiral link, preserving the fundamental symmetries of the spin quantum Hall effect. We demonstrate that, in the general case, the triplet sector of the theory remains coupled to the singlet sector. In the large-N limit, we systematically derive the effective long-distance, low-energy field theory, identified as a nonlinear sigma model. Our analysis reveals that while triplet modes are typically massive and do not influence the large-N nonlinear sigma model, specific conditions exist where these modes become soft, thereby increasing the diffusive length scale, which is the ultraviolet cutoff length of the effective theory. Furthermore, by calculating the bare longitudinal and spin Hall conductances, we show that the standard saddle-point approximation fails in regimes with significant tunneling asymmetry between even and odd links. Finally, we establish that the introduction of a Zeeman field not only breaks the SU(2) symmetry of the nonlinear sigma model action but also generates a term that explicitly violates inversion symmetry.

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References (98)

  1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  2. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  3. C.-K. Chiu, Jeffrey C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  4. E. P. Wigner, On a class of analytic functions from the quantum theory of collisions, Ann. Math. 53, 36 (1951).
  5. F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3, 140 (1962).
  6. F. J. Dyson, The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3, 1199 (1962).
  7. M. R. Zirnbauer, Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys. 37, 4986 (1996).
  8. A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997).
  9. P. Heinzner, A. Huckleberry, and M. R. Zirnbauer, Symmetry classes of disordered fermions, Commun. Math. Phys. 257, 725 (2005).
  10. A. P. Schnyder, S. Ryu, A. Furusaki, and Andreas W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  11. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors, AIP Conf. Proc. 1134, 10 (2009).
  12. A. Y. Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009).
  13. K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).
  14. D. C. Tsui and A. C. Gossard, Resistance standard using qunatization of the Hall resistance of GaAs-AlxGa1−xAs heterostructures, Appl. Phys. Lett. 38, 550 (1981).
  15. F. Evers, A. Mildenberger, and A. D. Mirlin, Multifractality at the quantum Hall transition: Beyond the parabolic paradigm, Phys. Rev. Lett. 101, 116803 (2008).
  16. M. R. Zirnbauer, Conformal field theory of the integer quantum Hall plateau transition, arXiv:hep-th/9905054.
  17. S. Kettemann and A. M. Tsvelik, Information about the integer quantum Hall transition extracted from the autocorrelation function of spectral determinants, Phys. Rev. Lett. 82, 3689 (1999).
  18. M. J. Bhaseen, I. I. Kogan, O. A. Soloviev, N. Taniguchi, and A. M. Tsvelik, Towards a field theory of the plateau transitions in the integer quantum Hall effect, Nucl. Phys. B 580, 688 (2000).
  19. A. M. Tsvelik, Wave functions statistics at quantum Hall critical point, arXiv:cond-mat/0112008.
  20. A. M. Tsvelik, Evidence for the PSL(2|2) Wess-Zumino-Novikov-Witten model as a model for the plateau transition in the quantum Hall effect: Evaluation of numerical simulations, Phys. Rev. B 75, 184201 (2007).
  21. R. Bondesan, D. Wieczorek, and M. Zirnbauer, Gaussian free fields at the integer quantum Hall plateau transition, Nucl. Phys. B 918, 52 (2017).
  22. M. R. Zirnbauer, The integer quantum Hall plateau transition is a current algebra after all, Nucl. Phys. B 941, 458 (2019).
  23. M. R. Zirnbauer, Marginal CFT perturbations at the integer quantum Hall transition, Ann. Phys. 431, 168559 (2021).
  24. M. R. Zirnbauer, On the infrared limit of the O(3) nonlinear σ-model at θ=π, arXiv:2408.12215.
  25. J. F. Karcher, N. Charles, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at spin quantum Hall transition, Ann. Phys. 435, 168584 (2021).
  26. J. Padayasi and I. Gruzberg, Conformal invariance and multifractality at Anderson transitions in arbitrary dimensions, Phys. Rev. Lett. 131, 266401 (2023).
  27. H. Obuse, A. R. Subramaniam, A. Furusaki, I. A. Gruzberg, and A. W. W. Ludwig, Boundary multifractality at the integer quantum Hall plateau transition: Implications for the critical theory, Phys. Rev. Lett. 101, 116802 (2008).
  28. 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).
  29. A. Klümper, W. Nuding, and A. Sedrakyan, Random network models with variable disorder of geometry, Phys. Rev. B 100, 140201(R) (2019).
  30. 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).
  31. E. J. Dresselhaus, B. Sbierski, and I. A. Gruzberg, Scaling collapse of longitudinal conductance near the integer quantum Hall transition, Phys. Rev. Lett. 129, 026801 (2022).
  32. H. Topchyan, I. A. Gruzberg, W. Nuding, A. Klümper, and A. Sedrakyan, The integer quantum Hall transition: An S-matrix approach to random networks, Phys. Rev. B 110, L081112 (2024).
  33. E. Macías, I. Gruzberg, and E. Bettelheim, Spin quantum Hall transition on random networks: Exact critical exponents via quantum gravity, arXiv:2601.22639.
  34. H. Levine, S. B. Libby, and Adrianus M. M. Pruisken, Electron delocalization by a magnetic field in two dimensions, Phys. Rev. Lett. 51, 1915 (1983).
  35. A. M. M. Pruisken, On localization in the theory of the quantized Hall effect: A two-dimensional realization of the θ-vacuum, Nucl. Phys. B 235, 277 (1984).
  36. D. Khmel'nitskii, Quantization of Hall conductivity, JETP Lett. 38, 552 (1983).
  37. A. M. M. Pruisken, Dilute instanton gas as the precursor of the integer quantum Hall effect, Phys. Rev. B 32, 2636 (1985).
  38. A. M. M. Pruisken, Quasiparticles in the theory of the integral quantum Hall effect (I), Nucl. Phys. B 285, 719 (1987).
  39. A. M. M. Pruisken, Quasiparticles in the theory of the integral quantum Hall effect (II). Renormalization of the Hall conductance or instanton angle theta, Nucl. Phys. B 290, 61 (1987).
  40. A. M. M. Pruisken and M. A. Baranov, Cracking Coulomb interactions in the quantum Hall regime, Europhys. Lett. 31, 543 (1995).
  41. A. Pruisken and I. Burmistrov, The instanton vacuum of generalized CPN−1 models, Ann. Phys. 316, 285 (2005).
  42. A. M. M. Pruisken and I. S. Burmistrov, θ renormalization, electron–electron interactions and super universality in the quantum Hall regime, Ann. Phys. 322, 1265 (2007).
  43. H. P. Wei, D. C. Tsui, M. A. Paalanen, and A. M. M. Pruisken, Experiments on delocalization and university in the integral quantum Hall effect, Phys. Rev. Lett. 61, 1294 (1988).
  44. S. Koch, R. J. Haug, K. v. Klitzing, and K. Ploog, Size-dependent analysis of the metal-insulator transition in the integral quantum Hall effect, Phys. Rev. Lett. 67, 883 (1991).
  45. R. T. F. van Schaijk, A. de Visser, S. M. Olsthoorn, H. P. Wei, and A. M. M. Pruisken, Probing the plateau-insulator quantum phase transition in the quantum Hall regime, Phys. Rev. Lett. 84, 1567 (2000).
  46. W. Li, G. A. Csáthy, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Scaling and universality of integer quantum Hall plateau-to-plateau transitions, Phys. Rev. Lett. 94, 206807 (2005).
  47. A. Pruisken, D. de Lang, L. Ponomarenko, and A. de Visser, Universal scaling results for the plateau–insulator transition in the quantum Hall regime, Solid State Commun. 137, 540 (2006).
  48. W. Li, C. L. Vicente, J. S. Xia, W. Pan, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Scaling in plateau-to-plateau transition: A direct connection of quantum Hall systems with the Anderson localization model, Phys. Rev. Lett. 102, 216801 (2009).
  49. W. Li, J. S. Xia, C. Vicente, N. S. Sullivan, W. Pan, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Crossover from the nonuniversal scaling regime to the universal scaling regime in quantum Hall plateau transitions, Phys. Rev. B 81, 033305 (2010).
  50. P. T. Madathil, K. A. Villegas Rosales, C. T. Tai, Y. J. Chung, L. N. Pfeiffer, K. W. West, K. W. Baldwin, and M. Shayegan, Delocalization and universality of the fractional quantum Hall plateau-to-plateau transitions, Phys. Rev. Lett. 130, 226503 (2023).
  51. S. Kaur, T. Chanda, K. R. Amin, D. Sahani, K. Watanabe, T. Taniguchi, U. Ghorai, Y. Gefen, G. J. Sreejith, and A. Bid, Universality of quantum phase transitions in the integer and fractional quantum Hall regimes, Nat. Commun. 15, 8535 (2024).
  52. C.-C. Yeh, P.-C. Liao, Y. Yang, W.-C. Lin, A. R. Panna, A. F. Rigosi, R. E. Elmquist, and C.-T. Liang, Conformity experiment on inelastic scattering exponent of electrons in two dimensions, Phys. Rev. Lett. 133, 096302 (2024).
  53. G. Volovik, On edge states in superconductors with time inversion symmetry breaking, Jetp Lett. 66, 522 (1997).
  54. V. Kagalovsky, B. Horovitz, Y. Avishai, and J. T. Chalker, Quantum Hall plateau transitions in disordered superconductors, Phys. Rev. Lett. 82, 3516 (1999).
  55. T. Senthil, J. B. Marston, and Matthew P. A. Fisher, Spin quantum Hall effect in unconventional superconductors, Phys. Rev. B 60, 4245 (1999).
  56. A similar relation between the spin current and the gradient of the magnetic field is realized in thin films of superfluid 3He-A [95, 96].
  57. F. Evers, Relaxation on critical percolation clusters, self-avoiding random walks, and the quantum Hall effect, Phys. Rev. E 55, 2321 (1997).
  58. I. A. Gruzberg, Andreas W. W. Ludwig, and N. Read, Exact exponents for the spin quantum Hall transition, Phys. Rev. Lett. 82, 4524 (1999).
  59. J. Cardy, Linking numbers for self-avoiding loops and percolation: Application to the spin quantum Hall transition, Phys. Rev. Lett. 84, 3507 (2000).
  60. E. J. Beamond, J. Cardy, and J. T. Chalker, Quantum and classical localization, the spin quantum Hall effect, and generalizations, Phys. Rev. B 65, 214301 (2002).
  61. A. D. Mirlin, F. Evers, and A. Mildenberger, Wavefunction statistics and multifractality at the spin quantum Hall transition, J. Phys. A: Math. Gen. 36, 3255 (2003).
  62. F. Evers, A. Mildenberger, and A. D. Mirlin, Multifractality at the spin quantum Hall transition, Phys. Rev. B 67, 041303(R) (2003).
  63. A. R. Subramaniam, I. A. Gruzberg, and Andreas W. W. Ludwig, Boundary criticality and multifractality at the two-dimensional spin quantum Hall transition, Phys. Rev. B 78, 245105 (2008).
  64. M. Puschmann, D. Hernangómez-Pérez, B. Lang, S. Bera, and F. Evers, Quartic multifractality and finite-size corrections at the spin quantum Hall transition, Phys. Rev. B 103, 235167 (2021).
  65. J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables, Phys. Rev. B 105, 184205 (2022).
  66. J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality at metal-insulator transitions and in metallic phases of two-dimensional disordered systems, Phys. Rev. B 106, 104202 (2022).
  67. J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Metal-insulator transition in a two-dimensional system of chiral unitary class, Phys. Rev. B 107, L020201 (2023).
  68. J. F. Karcher, I. A. Gruzberg, and A. D. Mirlin, Generalized multifractality in two-dimensional disordered systems of chiral symmetry classes, Phys. Rev. B 107, 104202 (2023).
  69. M. V. Parfenov and I. S. Burmistrov, Instanton analysis for the spin quantum Hall symmetry class: Nonperturbative corrections to physical observables and generalized multifractal spectrum, Phys. Rev. B 110, 165431 (2024).
  70. M. V. Parfenov and I. S. Burmistrov, Bulk-edge correspondence at the spin-to-integer quantum Hall effect crossover in topological superconductors, Phys. Rev. B 112, L161407 (2025).
  71. S. Y. F. Zhao, X. Cui, P. A. Volkov, H. Yoo, S. Lee, J. A. Gardener, A. J. Akey, R. Engelke, Y. Ronen, R. Zhong, G. Gu, S. Plugge, T. Tummuru, M. Kim, M. Franz, J. H. Pixley, N. Poccia, and P. Kim, Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors, Science 382, 1422 (2023).
  72. M. Martini, Y. Lee, T. Confalone, S. Shokri, C. N. Saggau, D. Wolf, G. Gu, K. Watanabe, T. Taniguchi, D. Montemurro, V. M. Vinokur, K. Nielsch, and N. Poccia, Twisted cuprate van der Waals heterostructures with controlled Josephson coupling, Mater. Today 67, 106 (2023).
  73. P. A. Volkov, J. H. Wilson, K. P. Lucht, and J. H. Pixley, Current- and field-induced topology in twisted nodal superconductors, Phys. Rev. Lett. 130, 186001 (2023).
  74. V. Pathak, O. Can, and M. Franz, Edge currents as probe of topology in twisted cuprate bilayers, Phys. Rev. B 110, 014506 (2024).
  75. M. V. Parfenov, V. S. Khrapai, and I. S. Burmistrov, Emergent spin quantum Hall edge states at the boundary of two-dimensional electron gas proximitized by an s-wave superconductor, arXiv:2605.05847.
  76. Z. Han, A. Allain, H. Arjmandi-Tash, K. Tikhonov, M. Feigel'man, B. Sacépé, and V. Bouchiat, Collapse of superconductivity in a hybrid tin–graphene Josephson junction array, Nat. Phys. 10, 380 (2014).
  77. J. T. Chalker and P. D. Coddington, Percolation, quantum tunneling and the integer quantum Hall effect, J. Phys. C: Solid State Phys. 21, 2665 (1988).
  78. K. Slevin and T. Ohtsuki, Finite size scaling of the Chalker-Coddington model, Int. J. Mod. Phys. Conf. Ser. 11, 60 (2012).
  79. M. R. Zirnbauer, Toward a theory of the integer quantum Hall transition: Continuum limit of the Chalker-Coddington model, J. Math. Phys. 38, 2007 (1997).
  80. A. Davis, Supersymmetry method for network models of quantum Hall transitions and hybrid structures, Ph.D. thesis, The Ohio State University, 2019, https://etd.ohiolink.edu/acprod/odb_etd/ws/send_file/send?accession=osu1566157715556164&Ydisposition=inline.
  81. D.-H. Lee, Network models of quantum percolation and their field-theory representations, Phys. Rev. B 50, 10788 (1994).
  82. The finite cutoff of Matsubara frequencies serves as an ultraviolet regularization of the Matsubara space, with indices restricted to |ɛn|≲1/τ, where τ is the elastic scattering time, cf. Eq. (30). This is the standard diffusive cutoff: the NLσM is applicable only at energies well below 1/τ, whereas higher-frequency modes are nondiffusive and merely renormalize the bare parameters of the action. Consequently, this truncation does not affect the long-distance, low-energy results, provided all external frequencies remain well below 1/τ. For a detailed discussion of this cutoff procedure, see Ref. [97].
  83. We note that one has to preserve the relation (7) between Ξ¯ and Ξ. Thus, the relation û=s2û*s2 should hold. Indeed, it satisfies for û∈SU(2).
  84. In the derivation of Eq. (20), we used the following Fierz identities involving σ={σ1,σ2,σ3}: trσσAσB=2trσAtrσB−trσAB and trσσAtrσσB=2trσAB−trσAtrσB, where the trace trσ operates in the spin space.
  85. T. Senthil and Matthew P. A. Fisher, Quasiparticle localization in superconductors with spin-orbit scattering, Phys. Rev. B 61, 9690 (2000).
  86. S. S. Babkin and I. S. Burmistrov, Generalized multifractality in the spin quantum Hall symmetry class with interaction, Phys. Rev. B 106, 125424 (2022).
  87. We note that there is also the contribution from the Jacobian for the part of Q̃j(0) that commutes with Λ [8]. However, this fact does not change the conclusion regarding the massive modes, which is similar to what happens in the iqHe [35].
  88. We note that these boundary conditions for the matrix Q are not general. For a detailed review on the boundary conditions in the qHe-type problems, see Refs. [69, 41].
  89. We note that the inequality Reβ̲(k)>0 guarantees that |β̲(k)|>0 and, consequently, the matrix β(0) has no zero eigenvalues. However, the absence of zero eigenvalues is possible even if Reβ̲(k)<0.
  90. S. Bhardwaj, I. A. Gruzberg, and V. Kagalovsky, Relevant perturbations at the spin quantum Hall transition, Phys. Rev. B 91, 035435 (2015).
  91. D. Bernard, N. Regnault, and D. Serban, Large N spin quantum Hall effect, Nucl. Phys. B 612, 291 (2001).
  92. M. Nadeem, M. S. Fuhrer, and X. Wang, The superconducting diode effect, Nat. Rev. Phys. 5, 558 (2023).
  93. In chiral symmetry classes, the Berry-phase weak topological term takes a familiar form; for a review, see, for example, Ref. [98].
  94. M. V. Feigel'man, A. I. Larkin, and M. A. Skvortsov, Keldysh action for disordered superconductors, Phys. Rev. B 61, 12361 (2000).
  95. G. E. Volovik and V. M. Yakovenko, Fractional charge, spin and statistics of solitons in superfluid He3 film, J. Phys.: Condens. Matter 1, 5263 (1989).
  96. G. E. Volovik, A. Solov'ev, and V. M. Yakovenko, Spin and statistics of soliton in a superfluid He3—A film, Pis'ma Zh. Eksp. Teor. Fiz. 49, 55 (1989).
  97. M. A. Baranov, A. M. M. Pruisken, and B. Škorić, (Mis-) handling gauge invariance in the theory of the quantum Hall effect. II. Perturbative results, Phys. Rev. B 60, 16821 (1999).
  98. P. Zhao, Z. Xiao, Y. Zhang, and R. Shindou, Topological effect on the Anderson transition in chiral symmetry classes, Phys. Rev. Lett. 133, 226601 (2024).

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Phys. Rev. B 114, 185405– Published 2 September, 2026
  • Received 12 April 2026
  • Revised 10 July 2026
  • Accepted 17 August 2026
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