- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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
  • Open Access

Critical Majorana Fermion at a Topological Quantum Hall Bilayer Transition

Cristian Voinea1, Wei Zhu2,3,*, Nicolas Regnault4,5,6, and Zlatko Papić1,†

  • *Contact author: zhuwei@westlake.edu.cn
  • †Contact author: z.papic@leeds.ac.uk
PDF

Phys. Rev. Lett. 136, 076601 – Published 20 February, 2026

DOI: https://doi.org/10.1103/mztz-fyk3

Abstract

Quantum Hall bilayers are a uniquely tunable platform that can realize continuous transitions between distinct topological phases of matter. One prominent example is the transition between the Halperin state and the Moore-Read Pfaffian, long predicted to host a critical theory of Majorana fermions but so far not verified in unbiased microscopic simulations. Using the fuzzy-sphere regularization, we identify the low-energy spectrum at this transition with the 3D gauged Majorana conformal field theory. We show that the transition is driven by the closing of the neutral fermion gap, and we directly extract the operator content in both integer and half-integer spin sectors. Our results resolve the long-standing question of the nature of a topological phase transition in a setting relevant to quantum Hall experiments, while also providing a realization of emergent fermionic fields on the fuzzy sphere, previously limited to bosonic fields.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (117)

  1. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  2. R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).
  3. J. Leinaas and J. Myrheim, On the theory of identical particles, Il Nuovo Cimento B Ser. 37, 1 (1977).
  4. F. Wilczek, Quantum mechanics of fractional-spin particles, Phys. Rev. Lett. 49, 957 (1982).
  5. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  6. J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, Cambridge, England, 2012).
  7. R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, Observation of an even-denominator quantum number in the fractional quantum Hall effect, Phys. Rev. Lett. 59, 1776 (1987).
  8. G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B360, 362 (1991).
  9. C. Nayak and F. Wilczek, 2n-quasihole states realize 2(n-1)-dimensional spinor braiding statistics in paired quantum Hall states, Nucl. Phys. B479, 529 (1996).
  10. N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, 10267 (2000).
  11. R. H. Morf, Transition from quantum Hall to compressible states in the second Landau level: New light on the ν=5/2 enigma, Phys. Rev. Lett. 80, 1505 (1998).
  12. E. H. Rezayi and F. D. M. Haldane, Incompressible paired Hall state, stripe order, and the composite fermion liquid phase in half-filled Landau levels, Phys. Rev. Lett. 84, 4685 (2000).
  13. M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Observation of half-integer thermal Hall conductance, Nature (London) 559, 205 (2018).
  14. A. A. Zibrov, C. R. Kometter, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level, Nature (London) 549, 360 (2017).
  15. J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, and C. R. Dean, Even denominator fractional quantum Hall states in bilayer graphene, Science 358, 648 (2017).
  16. A. A. Zibrov, E. M. Spanton, H. Zhou, C. Kometter, T. Taniguchi, K. Watanabe, and A. F. Young, Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene, Nat. Phys. 14, 930 (2018).
  17. J. Falson, D. Tabrea, D. Zhang, I. Sodemann, Y. Kozuka, A. Tsukazaki, M. Kawasaki, K. von Klitzing, and J. H. Smet, A cascade of phase transitions in an orbitally mixed half-filled landau level, Sci. Adv. 4, eaat8742 (2018).
  18. B. Dutta, W. Yang, R. Melcer, H. K. Kundu, M. Heiblum, V. Umansky, Y. Oreg, A. Stern, and D. Mross, Distinguishing between non-Abelian topological orders in a quantum Hall system, Science 375, 193 (2021).
  19. K. Huang, H. Fu, D. R. Hickey, N. Alem, X. Lin, K. Watanabe, T. Taniguchi, and J. Zhu, Valley isospin controlled fractional quantum Hall states in bilayer graphene, Phys. Rev. X 12, 031019 (2022).
  20. X. Liu, G. Farahi, C.-L. Chiu, Z. Papic, K. Watanabe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet, Science 375, 321 (2022).
  21. G. Farahi, C.-L. Chiu, X. Liu, Z. Papic, K. Watanabe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, Broken symmetries and excitation spectra of interacting electrons in partially filled Landau levels, Nat. Phys. 19, 1482 (2023).
  22. R. L. Willett, K. Shtengel, C. Nayak, L. N. Pfeiffer, Y. J. Chung, M. L. Peabody, K. W. Baldwin, and K. W. West, Interference measurements of non-abelian e/4 & Abelian e/2 quasiparticle braiding, Phys. Rev. X 13, 011028 (2023).
  23. Y. Hu, Y.-C. Tsui, M. He, U. Kamber, T. Wang, A. S. Mohammadi, K. Watanabe, T. Taniguchi, Z. Papić, M. P. Zaletel, and A. Yazdani, High-resolution tunnelling spectroscopy of fractional quantum Hall states, Nat. Phys. 21, 716 (2025).
  24. S. M. Girvin and A. H. MacDonald, Multicomponent quantum Hall systems: The sum of their parts and more, in Perspectives in Quantum Hall Effects (Wiley-VCH Verlag GmbH, New York, 2007), pp. 161–224.
  25. J. Eisenstein, Exciton condensation in bilayer quantum Hall systems, Annu. Rev. Condens. Matter Phys. 5, 159 (2014).
  26. Y. W. Suen, L. W. Engel, M. B. Santos, M. Shayegan, and D. C. Tsui, Observation of a ν=1/2 fractional quantum Hall state in a double-layer electron system, Phys. Rev. Lett. 68, 1379 (1992).
  27. J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and S. He, New fractional quantum Hall state in double-layer two-dimensional electron systems, Phys. Rev. Lett. 68, 1383 (1992).
  28. Y. W. Suen, H. C. Manoharan, X. Ying, M. B. Santos, and M. Shayegan, Origin of the ν=1/2 fractional quantum Hall state in wide single quantum wells, Phys. Rev. Lett. 72, 3405 (1994).
  29. J. Shabani, Y. Liu, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, Phase diagrams for the stability of the ν=12 fractional quantum Hall effect in electron systems confined to symmetric, wide GaAs quantum wells, Phys. Rev. B 88, 245413 (2013).
  30. S. K. Singh, C. Wang, C. T. Tai, C. S. Calhoun, K. A. Villegas Rosales, P. T. Madathil, A. Gupta, K. W. Baldwin, L. N. Pfeiffer, and M. Shayegan, Topological phase transition between Jain states and daughter states of the ν=1/2 fractional quantum Hall state, Nat. Phys. 20, 1247 (2024).
  31. S. K. Singh, C. Wang, A. Gupta, K. W. Baldwin, L. N. Pfeiffer, and M. Shayegan, Fractional quantum Hall state at ν=1/2 with energy gap up to 6 K, and possible transition from one- to two-component state, Phys. Rev. Lett. 135, 246603 (2025).
  32. B. I. Halperin, Theory of the quantized Hall conductance, Helv. Phys. Acta 56, 75 (1983).
  33. S. He, S. Das Sarma, and X. C. Xie, Quantized Hall effect and quantum phase transitions in coupled two-layer electron systems, Phys. Rev. B 47, 4394 (1993).
  34. K. Nomura and D. Yoshioka, Gap evolution in ν=1/2 bilayer quantum Hall systems, J. Phys. Soc. Jpn. 73, 2612 (2004).
  35. Z. Papić, M. O. Goerbig, N. Regnault, and M. V. Milovanović, Tunneling-driven breakdown of the 331 state and the emergent Pfaffian and composite Fermi liquid phases, Phys. Rev. B 82, 075302 (2010).
  36. M. R. Peterson and S. Das Sarma, Quantum hall phase diagram of half-filled bilayers in the lowest and the second orbital landau levels: Abelian versus non-Abelian incompressible fractional quantum Hall states, Phys. Rev. B 81, 165304 (2010).
  37. M. R. Peterson, Z. Papić, and S. Das Sarma, Fractional quantum Hall effects in bilayers in the presence of interlayer tunneling and charge imbalance, Phys. Rev. B 82, 235312 (2010).
  38. Z. Liu, A. Vaezi, C. Repellin, and N. Regnault, Phase diagram of ν=12+12 bilayer bosons with interlayer couplings, Phys. Rev. B 93, 085115 (2016).
  39. W. Zhu, Z. Liu, F. D. M. Haldane, and D. N. Sheng, Fractional quantum Hall bilayers at half filling: Tunneling-driven non-Abelian phase, Phys. Rev. B 94, 245147 (2016).
  40. D. Cabra, A. Lopez, and G. Rossini, Transition from abelian to non-Abelian FQHE states, Eur. Phys. J. B 19, 21 (2001).
  41. A. Cappelli, L. S. Georgiev, and I. T. Todorov, Parafermion hall states from coset projections of Abelian conformal theories, Nucl. Phys. B599, 499 (2001).
  42. C. Repellin, T. Neupert, B. A. Bernevig, and N. Regnault, Projective construction of the ℤk Read-Rezayi fractional quantum Hall states and their excitations on the torus geometry, Phys. Rev. B 92, 115128 (2015).
  43. V. Crépel, B. Estienne, and N. Regnault, Variational ansatz for an Abelian to non-Abelian topological phase transition in ν=1/2+1/2 bilayers, Phys. Rev. Lett. 123, 126804 (2019).
  44. J. K. Jain, Composite-fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63, 199 (1989).
  45. X.-G. Wen, Continuous topological phase transitions between clean quantum Hall states, Phys. Rev. Lett. 84, 3950 (2000).
  46. M. Barkeshli and X.-G. Wen, Anyon condensation and continuous topological phase transitions in non-Abelian fractional quantum Hall states, Phys. Rev. Lett. 105, 216804 (2010).
  47. M. Barkeshli and X.-G. Wen, Bilayer quantum hall phase transitions and the orbifold non-Abelian fractional quantum Hall states, Phys. Rev. B 84, 115121 (2011).
  48. S. R. Elliott and M. Franz, Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys. 87, 137 (2015).
  49. A. Yazdani, F. von Oppen, B. I. Halperin, and A. Yacoby, Hunting for Majoranas, Science 380, eade0850 (2023).
  50. W. Zhu, C. Han, E. Huffman, J. S. Hofmann, and Y.-C. He, Uncovering Conformal Symmetry in the 3D Ising transition: State-operator correspondence from a quantum fuzzy sphere regularization, Phys. Rev. X 13, 021009 (2023).
  51. P. D. Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer, New York, 1997).
  52. J. L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17, L385 (1984).
  53. J. L. Cardy, Universal amplitudes in finite-size scaling: Generalisation to arbitrary dimensionality, J. Phys. A 18, L757 (1985).
  54. F. D. M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51, 605 (1983).
  55. L. Hu, Y.-C. He, and W. Zhu, Operator product expansion coefficients of the 3D Ising criticality via quantum fuzzy spheres, Phys. Rev. Lett. 131, 031601 (2023).
  56. L. Hu, W. Zhu, and Y.-C. He, Entropic f function of three-dimensional Ising conformal field theory via fuzzy sphere regularization, Phys. Rev. B 111, 155151 (2025).
  57. C. Han, L. Hu, W. Zhu, and Y.-C. He, Conformal four-point correlators of the three-dimensional Ising transition via the quantum fuzzy sphere, Phys. Rev. B 108, 235123 (2023).
  58. J. S. Hofmann, F. Goth, W. Zhu, Y.-C. He, and E. Huffman, Quantum Monte Carlo simulation of the 3D Ising transition on the fuzzy sphere, SciPost Phys. Core 7, 028 (2024).
  59. Z. Zhou and Y. Zou, Studying the 3d Ising surface CFTs on the fuzzy sphere, SciPost Phys. 18, 031 (2025).
  60. G. Fardelli, A. L. Fitzpatrick, and E. Katz, Constructing the infrared conformal generators on the fuzzy sphere, SciPost Phys. 18, 086 (2025).
  61. C. Han, L. Hu, and W. Zhu, Conformal operator content of the Wilson-Fisher transition on fuzzy sphere bilayers, Phys. Rev. B 110, 115113 (2024).
  62. R. Fan, J. Dong, and A. Vishwanath, Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere, arXiv:2505.06342.
  63. E. A. Cruz, I. R. Klebanov, G. Tarnopolsky, and Y. Xin, Yang-Lee quantum criticality in various dimensions, arXiv:2505.06369.
  64. J. E. Miro and O. Delouche, Flowing from the Ising model on the fuzzy sphere to the 3D Lee-Yang CFT, J. High Energy Phys. 10 (2025) 037.
  65. Z. Zhou and Y.-C. He, 3D conformal field theories with Sp(n) global symmetry on a fuzzy sphere, Phys. Rev. Lett. 135, 026504 (2025).
  66. Z. Zhou, L. Hu, W. Zhu, and Y.-C. He, SO(5) deconfined phase transition under the fuzzy-sphere microscope: Approximate conformal symmetry, pseudo-criticality, and operator spectrum, Phys. Rev. X 14, 021044 (2024).
  67. S. Yang, L. dong Hu, C. Han, W. Zhu, and Y. Chen, Conformal operator flows of the deconfined quantum criticality from SO(5) to O(4), arXiv:2507.01322.
  68. Z. Zhou, C. Wang, and Y.-C. He, Chern-Simons-matter conformal field theory on fuzzy sphere: Confinement transition of Kalmeyer-Laughlin chiral spin liquid, arXiv:2507.19580.
  69. J. Taylor, C. Voinea, Z. Papić, and R. Fan, Conformal scalar field theory from Ising tricriticality on the fuzzy sphere, arXiv:2506.22539.
  70. Y.-C. He, Free real scalar CFT on fuzzy sphere: Spectrum, algebra and wavefunction ansatz, arXiv:2506.14904.
  71. C. Voinea, R. Fan, N. Regnault, and Z. Papić, Regularizing 3D conformal field theories via anyons on the fuzzy sphere, Phys. Rev. X 15, 031007 (2025).
  72. J. C. Teo, T. L. Hughes, and E. Fradkin, Theory of twist liquids: Gauging an anyonic symmetry, Ann. Phys. (Amsterdam) 360, 349 (2015).
  73. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mztz-fyk3 for details of the derivations and further numerical results, which contains Refs. [74–89].
  74. E. H. Fradkin, C. Nayak, A. Tsvelik, and F. Wilczek, A Chern-Simons effective field theory for the Pfaffian quantum Hall state, Nucl. Phys. B516, 704 (1998).
  75. E. Fradkin, C. Nayak, and K. Schoutens, Landau-Ginzburg theories for non-Abelian quantum Hall states, Nucl. Phys. B546, 711 (1999).
  76. M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100, 115147 (2019).
  77. P.-S. Hsin and N. Seiberg, Level/rank duality and Chern-Simons-Matter theories, J. High Energy Phys. 09 (2016) 095.
  78. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351 (1989).
  79. J. C. Y. Teo and C. L. Kane, From Luttinger liquid to non-Abelian quantum Hall states, Phys. Rev. B 89, 085101 (2014).
  80. G. Möller, A. Wójs, and N. R. Cooper, Neutral fermion excitations in the Moore-Read state at filling factor ν=5/2, Phys. Rev. Lett. 107, 036803 (2011).
  81. P. Bonderson, A. E. Feiguin, and C. Nayak, Numerical calculation of the neutral fermion gap at the ν=5/2 fractional quantum Hall state, Phys. Rev. Lett. 106, 186802 (2011).
  82. S. R. Renn and B. W. Roberts, Magnetorotons and the fractional quantum Hall effect in double-quantum-well systems, Phys. Rev. B 48, 10926 (1993).
  83. A. H. MacDonald and S.-C. Zhang, Collective excitations in double-layer quantum hall systems, Phys. Rev. B 49, 17208 (1994).
  84. K. Shizuya, Single-mode approximation and effective Chern–Simons theories for quantum Hall systems, Int. J. Mod. Phys. B 17, 5875 (2003).
  85. S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Magneto-Roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B 33, 2481 (1986).
  86. R. H. Morf, N. d’Ambrumenil, and S. Das Sarma, Excitation gaps in fractional quantum Hall states: An exact diagonalization study, Phys. Rev. B 66, 075408 (2002).
  87. J. K. Jain and R. K. Kamilla, Composite fermions in the Hilbert space of the lowest electronic Landau level, Int. J. Mod. Phys. B 11, 2621 (1997).
  88. A. C. Balram and A. Wójs, Fractional quantum Hall effect at ν=2+4/9, Phys. Rev. Res. 2, 032035 (2020).
  89. N. Read and E. Rezayi, Quasiholes and fermionic zero modes of paired fractional quantum Hall states: The mechanism for non-Abelian statistics, Phys. Rev. B 54, 16864 (1996).
  90. X. G. Wen and A. Zee, Shift and spin vector: New topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69, 953 (1992).
  91. T.-L. Ho, Broken symmetry of two-component ν=1/2 quantum Hall states, Phys. Rev. Lett. 75, 1186 (1995).
  92. N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, Quantum phases of vortices in rotating Bose-Einstein condensates, Phys. Rev. Lett. 87, 120405 (2001).
  93. N. Regnault and T. Jolicoeur, Quantum Hall fractions in rotating Bose-Einstein condensates, Phys. Rev. Lett. 91, 030402 (2003).
  94. H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  95. J. Dubail, N. Read, and E. H. Rezayi, Real-space entanglement spectrum of quantum Hall systems, Phys. Rev. B 85, 115321 (2012).
  96. A. Sterdyniak, A. Chandran, N. Regnault, B. A. Bernevig, and P. Bonderson, Real-space entanglement spectrum of quantum Hall states, Phys. Rev. B 85, 125308 (2012).
  97. X.-G. Wen, Topological order and edge structure of ν=1/2 quantum Hall state, Phys. Rev. Lett. 70, 355 (1993).
  98. M. Milovanović and N. Read, Edge excitations of paired fractional quantum Hall states, Phys. Rev. B 53, 13559 (1996).
  99. B.-X. Lao and S. Rychkov, 3D Ising CFT and exact diagonalization on icosahedron: The power of conformal perturbation theory, SciPost Phys. 15, 243 (2023).
  100. A. M. Läuchli, L. Herviou, P. H. Wilhelm, and S. Rychkov, Exact diagonalization, matrix product states and conformal perturbation theory study of a 3D Ising fuzzy sphere model, SciPost Phys. 19, 076 (2025).
  101. C. Wang and T. Senthil, Composite fermi liquids in the lowest landau level, Phys. Rev. B 94, 245107 (2016).
  102. S. D. Geraedts, C. Repellin, C. Wang, R. S. K. Mong, T. Senthil, and N. Regnault, Emergent particle-hole symmetry in spinful bosonic quantum Hall systems, Phys. Rev. B 96, 075148 (2017).
  103. J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum Hall state with ultracold atoms, Nature (London) 619, 495 (2023).
  104. C. Wang, F.-M. Liu, M.-C. Chen, H. Chen, X.-H. Zhao, C. Ying, Z.-X. Shang, J.-W. Wang, Y.-H. Huo, C.-Z. Peng, X. Zhu, C.-Y. Lu, and J.-W. Pan, Realization of fractional quantum Hall state with interacting photons, Science 384, 579 (2024).
  105. J. Liang, Z. Liu, Z. Yang, Y. Huang, U. Wurstbauer, C. R. Dean, K. W. West, L. N. Pfeiffer, L. Du, and A. Pinczuk, Evidence for chiral graviton modes in fractional quantum Hall liquids, Nature (London) 628, 78 (2024).
  106. I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Resonantly enhanced tunneling in a double layer quantum Hall ferromagnet, Phys. Rev. Lett. 84, 5808 (2000).
  107. Y.-H. Zhang, Z. Zhu, and A. Vishwanath, XY* transition and extraordinary boundary criticality from fractional exciton condensation in quantum Hall bilayer, Phys. Rev. X 13, 031023 (2023).
  108. M. Barkeshli and J. McGreevy, Continuous transition between fractional quantum hall and superfluid states, Phys. Rev. B 89, 235116 (2014).
  109. K. Yang, Interface and phase transition between Moore-Read and Halperin 331 fractional quantum Hall states: Realization of chiral Majorana fermion, Phys. Rev. B 96, 241305 (2017).
  110. K. K. W. Ma and K. Yang, Simple analog of the black-hole information paradox in quantum Hall interfaces, Phys. Rev. B 105, 045306 (2022).
  111. V. Crépel, B. Estienne, and N. Regnault, Variational ansatz for an Abelian to Non-Abelian topological phase transition in ν=1/2+1/2 bilayers, Phys. Rev. Lett. 123, 126804 (2019).
  112. L. Fei, S. Giombi, I. R. Klebanov, and G. Tarnopolsky, Yukawa conformal field theories and emergent supersymmetry, Prog. Theor. Exp. Phys. 2016, 12C105 (2016).
  113. Z.-Q. Gao, T. Wang, and D.-H. Lee, Interacting Chern insulator transition on the sphere: Revealing the Gross-Neveu-Yukawa criticality, arXiv:2504.15338.
  114. Z. Zhou, D. Gaiotto, and Y.-C. He, Free majorana fermion meets gauged ising conformal field theory on the fuzzy sphere, arXiv:2509.08038.
  115. DiagHamdiagham, https://www.nick-ux.org/diagham.
  116. Z. Zhou, fuzzified—julia package for numerics on the fuzzy sphere, arXiv:2503.00100.
  117. Cristian Voinea, Wei Zhu, Nicolas Regnault, and Zlatko Papić, 10.5518/1817.

Outline

Information

Phys. Rev. Lett. 136, 076601– Published 20 February, 2026
  • Received 2 October 2025
  • Accepted 26 January 2026
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