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Conformal Scalar Field Theory from Ising Tricriticality on the Fuzzy Sphere

Joseph Taylor1, Cristian Voinea1, Zlatko Papić1, and Ruihua Fan2

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Phys. Rev. Lett. 136, 056503 – Published 6 February, 2026

DOI: https://doi.org/10.1103/cj3l-cf58

Abstract

Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories. Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point—a nontrivial fixed point where first-order and second-order transitions meet—which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT. Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories—previously inaccessible—can emerge from interacting electrons in this framework.

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References (48)

  1. S. Rychkov, EPFL Lectures on Conformal Field Theory in D >= 3 Dimensions, SpringerBriefs in Physics (Springer, Cham, 2016).
  2. D. Simmons-Duffin, The conformal bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (World Scientific, Singapore, 2017), pp. 1–74.
  3. D. Poland, S. Rychkov, and A. Vichi, The conformal bootstrap: Theory, numerical techniques, and applications, Rev. Mod. Phys. 91, 015002 (2019).
  4. J. L. Cardy, Conformal invariance and universality in finite-size scaling, J. Phys. A 17, L385 (1984).
  5. The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer-Verlag, New York, 1987).
  6. 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).
  7. 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).
  8. 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).
  9. G. Fardelli, A. L. Fitzpatrick, and E. Katz, Constructing the infrared conformal generators on the fuzzy sphere, SciPost Phys. 18, 086 (2025).
  10. R. Fan, Note on explicit construction of conformal generators on the fuzzy sphere, arXiv:2409.08257.
  11. 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).
  12. L. Hu, Y.-C. He, and W. Zhu, Solving conformal defects in 3D conformal field theory using fuzzy sphere regularization, Nat. Commun. 15, 3659 (2024).
  13. Z. Zhou, D. Gaiotto, Y.-C. He, and Y. Zou, The g-function and defect changing operators from wavefunction overlap on a fuzzy sphere, SciPost Phys. 17, 021 (2024).
  14. Z. Zhou and Y. Zou, Studying the 3D Ising surface CFTs on the fuzzy sphere, SciPost Phys. 18, 031 (2025).
  15. M. Dedushenko, Ising BCFT from fuzzy hemisphere, arXiv:2407.15948.
  16. 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).
  17. 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).
  18. 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).
  19. M. Ippoliti, Roger S. K. Mong, F. F. Assaad, and M. P. Zaletel, Half-filled Landau levels: A continuum and sign-free regularization for three-dimensional quantum critical points, Phys. Rev. B 98, 235108 (2018).
  20. 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).
  21. B.-B. Chen, X. Zhang, Y. Wang, K. Sun, and Z. Y. Meng, Phases of (2+1)D SO(5) nonlinear sigma model with a topological term on a sphere: Multicritical point and disorder phase, Phys. Rev. Lett. 132, 246503 (2024).
  22. B.-B. Chen, X. Zhang, and Z. Y. Meng, Emergent conformal symmetry at the multicritical point of (2+1)D SO(5) model with Wess-Zumino-Witten term on a sphere, Phys. Rev. B 110, 125153 (2024).
  23. S. Yang, Y.-G. Yue, Y. Tang, C. Han, W. Zhu, and Y. Chen, Microscopic study of 3D Potts phase transition via fuzzy sphere regularization, Phys. Rev. B 112, 024436 (2025).
  24. R. Fan, J. Dong, and A. Vishwanath, Simulating the non-unitary Yang-Lee conformal field theory on the fuzzy sphere, arXiv:2505.06342.
  25. E. A. Cruz, I. R. Klebanov, G. Tarnopolsky, and Y. Xin, Yang-Lee quantum criticality in various dimensions, arXiv:2505.06369.
  26. 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.
  27. 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).
  28. J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, England, 1996), Vol. 5.
  29. J. Henriksson, The tricritical Ising CFT and conformal bootstrap, J. High Energy Phys. 08 (2025) 031.
  30. M. Blume, Theory of the first-order magnetic phase change in UO2, Phys. Rev. 141, 517 (1966).
  31. H. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting, Physica (Utrecht) 32, 966 (1966).
  32. F. D. M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid states, Phys. Rev. Lett. 51, 605 (1983).
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/cj3l-cf58 for details of the derivations and further numerical results.
  34. T. Grover, D. N. Sheng, and A. Vishwanath, Emergent space-time supersymmetry at the boundary of a topological phase, Science 344, 280 (2014).
  35. A. Rahmani, X. Zhu, M. Franz, and I. Affleck, Emergent supersymmetry from strongly interacting Majorana zero modes, Phys. Rev. Lett. 115, 166401 (2015).
  36. E. O’Brien and P. Fendley, Lattice supersymmetry and order-disorder coexistence in the tricritical Ising model, Phys. Rev. Lett. 120, 206403 (2018).
  37. K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Spontaneous interlayer coherence in double-layer quantum Hall systems: Charged vortices and Kosterlitz-Thouless phase transitions, Phys. Rev. B 51, 5138 (1995).
  38. B. I. Halperin, P. A. Lee, and N. Read, Theory of the half-filled Landau level, Phys. Rev. B 47, 7312 (1993).
  39. 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).
  40. P. M. Chaikin, T. C. Lubensky, and T. A. Witten, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, England, 1995), Vol. 10.
  41. 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).
  42. A. Cappelli, C. A. Trugenberger, and G. R. Zemba, Infinite symmetry in the quantum Hall effect, Nucl. Phys. B396, 465 (1993).
  43. A. Gromov and D. T. Son, Bimetric theory of fractional quantum Hall states, Phys. Rev. X 7, 041032 (2017).
  44. 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).
  45. diagham, https://www.nick-ux.org/diagham.
  46. Z. Zhou, fuzzified—julia package for numerics on the fuzzy sphere, arXiv:2503.00100.
  47. Y.-C. He, Free real scalar CFT on fuzzy sphere: Spectrum, algebra and wavefunction ansatz, arXiv:2506.14904.
  48. J. Taylor, C. Voinea, Z. Papić, and R. Fan, 10.5518/1789.

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Phys. Rev. Lett. 136, 056503– Published 6 February, 2026
  • Received 7 July 2025
  • Accepted 24 December 2025
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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.

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