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Spontaneously Broken Noninvertible Symmetries in Transverse-Field Ising Qudit Chains

Kristian Tyn Kai Chung1,2,*, Umberto Borla3,4,5, Andriy H. Nevidomskyy2,6,7, and Sergej Moroz8,9

  • *Contact author: ktchung.phys@gmail.com
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Phys. Rev. Lett. 136, 216601 – Published 27 May, 2026

DOI: https://doi.org/10.1103/ng8b-sdt4

Abstract

Recent developments have revealed that symmetries need not form a group, but instead can be noninvertible. Here we use analytical arguments and numerical evidence to illuminate how spontaneous symmetry breaking of a noninvertible symmetry is similar yet distinct from ordinary, invertible, symmetry breaking. We consider one-dimensional chains of group-valued qudits, whose local Hilbert space is spanned by elements of a finite group G (reducing to ordinary qubits when G=Z2). We construct Ising-type transverse-field Hamiltonians with Rep(G) symmetry whose generators multiply according to the tensor product of irreducible representations (irreps) of the group G. For non-Abelian G, the symmetry is noninvertible. In the symmetry broken phase there is one ground state per irrep on a closed chain. The symmetry breaking can be detected by local order parameters but, unlike the invertible case, different ground states have distinct entanglement patterns. We show that for each irrep of dimension greater than one the corresponding ground state exhibits string order, entanglement spectrum degeneracies, and has gapless edge modes on an open chain—features usually associated with symmetry-protected topological order. Consequently, domain wall excitations behave as one-dimensional non-Abelian anyons with nontrivial internal Hilbert spaces and fusion rules. Our Letter identifies properties of noninvertible symmetry breaking that existing quantum hardware can probe.

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References (83)

  1. L. Landau, The theory of phase transitions, Nature (London) 138, 840 (1936).
  2. E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. (N.Y.) 16, 407 (1961).
  3. R. Peierls, On Ising’s model of ferromagnetism, Math. Proc. Cambridge Philos. Soc. 32, 477 (1936).
  4. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  5. C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85, 808 (1952).
  6. P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys. (N.Y.) 4, 287 (1963).
  7. T. Senthil, Symmetry-protected topological phases of quantum matter, Annu. Rev. Condens. Matter Phys. 6, 299 (2015).
  8. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (Springer, New York, 2019), 10.1007/978-1-4939-9084-9.
  9. M. den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709 (1989).
  10. T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S=1 quantum spin chains, Commun. Math. Phys. 147, 431 (1992).
  11. F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012).
  12. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  13. D. V. Else, S. D. Bartlett, and A. C. Doherty, Hidden symmetry-breaking picture of symmetry-protected topological order, Phys. Rev. B 88, 085114 (2013).
  14. R. Verresen, R. Moessner, and F. Pollmann, One-dimensional symmetry protected topological phases and their transitions, Phys. Rev. B 96, 165124 (2017).
  15. D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  16. J. McGreevy, Generalized symmetries in condensed matter, Annu. Rev. Condens. Matter Phys. 14, 57 (2023).
  17. V. B. Petkova and J.-B. Zuber, Generalised twisted partition functions, Phys. Lett. B 504, 157 (2001).
  18. J. Fröhlich, J. Fuchs, I. Runkel, and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93, 070601 (2004).
  19. J. Fröhlich, J. Fuchs, I. Runkel, and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B763, 354 (2007).
  20. D. Aasen, R. S. Mong, and P. Fendley, Topological defects on the lattice: I. The Ising model, J. Phys. A 49, 354001 (2016).
  21. D. Aasen, P. Fendley, and R. S. K. Mong, Topological defects on the lattice: Dualities and degeneracies, arXiv:2008.08598.
  22. S.-H. Shao, What’s done cannot be undone: TASI lectures on non-invertible symmetry, arXiv:2308.00747.
  23. S. Schafer-Nameki, ICTP lectures on (non-)invertible generalized symmetries, arXiv:2305.18296.
  24. L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, Categorical Landau paradigm for gapped phases, Phys. Rev. Lett. 133, 161601 (2024).
  25. C. Cordova, T. T. Dumitrescu, K. Intriligator, and S.-H. Shao, Snowmass white paper: Generalized symmetries in quantum field theory and beyond, arXiv:2205.09545.
  26. L. Bhardwaj, S. Schäfer-Nameki, and A. Tiwari, Unifying constructions of non-invertible symmetries, SciPost Phys. 15, 122 (2023).
  27. L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, The club sandwich: Gapless phases and phase transitions with non-invertible symmetries, SciPost Phys. 18, 156 (2025).
  28. K. Roumpedakis, S. Seifnashri, and S.-H. Shao, Higher gauging and non-invertible condensation defects, Commun. Math. Phys. 401, 3043 (2023).
  29. L. Bhardwaj and S. Schäfer-Nameki, Generalized charges, part II: Non-invertible symmetries and the symmetry TFT, SciPost Phys. 19, 098 (2025).
  30. L. Bhardwaj, L. E. Bottini, D. Pajer, and S. Schäfer-Nameki, Gapped phases with non-invertible symmetries: (1+1)d, SciPost Phys. 18, 032 (2025).
  31. P.-S. Hsin, R. Kobayashi, and C. Zhang, Fractionalization of coset non-invertible symmetry and exotic Hall conductance, SciPost Phys. 17, 095 (2024).
  32. L. Bhardwaj, D. Pajer, S. Schäfer-Nameki, and A. Warman, Hasse diagrams for gapless SPT and SSB phases with non-invertible symmetries, SciPost Phys. 19, 113 (2025).
  33. T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, Non-invertible symmetries and higher representation theory I, SciPost Phys. 17, 015 (2024).
  34. L. Bhardwaj, L. E. Bottini, S. Schafer-Nameki, and A. Tiwari, Lattice models for phases and transitions with non-invertible symmetries, arXiv:2405.05964.
  35. L. Bhardwaj, L. E. Bottini, S. Schäfer-Nameki, and A. Tiwari, Illustrating the categorical Landau paradigm in lattice models, Phys. Rev. B 111, 054432 (2025).
  36. A. Chatterjee, Ömer M. Aksoy, and X.-G. Wen, Quantum phases and transitions in spin chains with non-invertible symmetries, SciPost Phys. 17, 115 (2024).
  37. L. Lootens, C. Delcamp, and F. Verstraete, Entanglement and the density matrix renormalization group in the generalized Landau paradigm, Nat. Phys. 21, 1657 (2025).
  38. R. Kobayashi and H. Watanabe, Projective representations, bogomolov multiplier, and their applications in physics, Int. J. Theor. Phys. 65, 65 (2026).
  39. L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Symmetric Hamiltonians and matrix product operator intertwiners, PRX Quantum 4, 020357 (2023).
  40. L. Lootens, C. Delcamp, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Topological sectors, PRX Quantum 5, 010338 (2024).
  41. S. D. Pace, A. Chatterjee, and S.-H. Shao, Lattice T-duality from non-invertible symmetries in quantum spin chains, SciPost Phys. 18, 121 (2025).
  42. Y. Choi, Y. Sanghavi, S.-H. Shao, and Y. Zheng, Non-invertible and higher-form symmetries in 2+1d lattice gauge theories, SciPost Phys. 18, 008 (2025).
  43. P. Gorantla, S.-H. Shao, and N. Tantivasadakarn, Tensor networks for noninvertible symmetries in 3+1D and beyond, Phys. Rev. X 15, 041006 (2025).
  44. N. Seiberg, S. Seifnashri, and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space, SciPost Phys. 16, 154 (2024).
  45. S. Seifnashri and S.-H. Shao, Cluster state as a noninvertible symmetry-protected topological phase, Phys. Rev. Lett. 133, 116601 (2024).
  46. D.-C. Lu, Z. Sun, and Y.-Z. You, Realizing triality and p-ality by lattice twisted gauging in (1+1)d quantum spin systems, SciPost Phys. 17, 136 (2024).
  47. W. Cao, Y. Miao, and M. Yamazaki, Global symmetries of quantum lattice models under non-invertible dualities, SciPost Phys. Core 8, 070 (2025).
  48. L. Eck and P. Fendley, From the XXZ chain to the integrable Rydberg-blockade ladder via non-invertible duality defects, SciPost Phys. 16, 127 (2024).
  49. S. D. Pace, G. Delfino, H. T. Lam, and Ö. M. Aksoy, Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections, SciPost Phys. 18, 021 (2025).
  50. W. Cao, L. Li, and M. Yamazaki, Generating lattice non-invertible symmetries, SciPost Phys. 17, 104 (2024).
  51. C. Fechisin, N. Tantivasadakarn, and V. V. Albert, Noninvertible symmetry-protected topological order in a group-based cluster state, Phys. Rev. X 15, 011058 (2025).
  52. J. Graf, F. M. Surace, M. Berg, and S. Moroz, The Ising dual-reflection interface: Z4 symmetry, Majorana strong zero modes and SPT phases, arXiv:2412.06377.
  53. S. D. Pace, H. T. Lam, and Ömer M. Aksoy, (SPT-)LSM theorems from projective non-invertible symmetries, SciPost Phys. 18, 028 (2025).
  54. A. Warman, F. Yang, A. Tiwari, H. Pichler, and S. Schäfer-Nameki, Categorical symmetries in spin models with atom arrays, Phys. Rev. Lett. 135, 206503 (2025).
  55. D.-C. Lu, F. Xu, and Y.-Z. You, Strange correlator and string order parameter for non-invertible symmetry protected topological phases in (1+1)d, arXiv:2505.00673.
  56. E. Fradkin and L. P. Kadanoff, Disorder variables and para-fermions in two-dimensional statistical mechanics, Nucl. Phys. B170, 1 (1980).
  57. P. Fendley, Parafermionic edge zero modes in Zn-invariant spin chains, J. Stat. Mech. (2012) P11020.
  58. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ng8b-sdt4 for details on G-qudits, exact SSB ground states at h=0, multiplet structure for noninvertible symmetry and relations between local and nonlocal order parameters.
  59. C. G. Brell, Generalized cluster states based on finite groups, New J. Phys. 17, 023029 (2015).
  60. V. V. Albert, D. Aasen, W. Xu, W. Ji, J. Alicea, and J. Preskill, Spin chains, defects, and quantum wires for the quantum-double edge, arXiv:2111.12096.
  61. S. D. Pace, H. T. Lam, and Ö. M. Aksoy, (SPT-)LSM theorems from projective non-invertible symmetries, SciPost Phys. 18, 028 (2025).
  62. A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (Amsterdam) 303, 2 (2003).
  63. Morten I. K. Munk, A. Rasmussen, and M. Burrello, Dyonic zero-energy modes, Phys. Rev. B 98, 245135 (2018).
  64. B. V.-D. Cuiper, W. Wiesiolek, and F. Verstraete, Les Houches lecture notes on tensor networks, arXiv:2512.24390.

  65. A technical note regarding language: linear combinations of the RΓ operators form a fusion algebra, which may contain formal inverses, i.e., some linear combination xΓa=ΣbcbRΓb such that xΓaRΓa=1. The noninvertibility of the symmetry refers instead to treating the operators as forming a representation of the fusion category Rep(G), for which Eq. (8) encodes the fusion of simple objects.

  66. The reader may notice that this phase transition occurs precisely at h=J. This is a consequence of the fine-tuned nature of Eq. (6): it is dual to Eq. (5), which can be written in the form HG=−J∑iδ(gi,gi+1)−h∑i|+⟩i⟨+|i, where we define the equal superposition state |+⟩=|G|−1/2∑g|g⟩. This Hamiltonian is actually a |G|-state Potts model, which is self-dual [67] with a first-order transition if |G|≥4 [68]. Generic Hamiltonians Eqs. (a1) and (a2), can exhibit continuous or discontinuous transitions depending on the parameters.

  67. J. Sólyom and P. Pfeuty, Renormalization-group study of the Hamiltonian version of the Potts model, Phys. Rev. B 24, 218 (1981).
  68. F. Iglói and J. Sólyom, First-order transition for the (1+1)-dimensional q≥4 Potts model from finite lattice extrapolation, J. Phys. C 16, 2833 (1983).
  69. J. Alicea and P. Fendley, Topological phases with parafermions: Theory and blueprints, Annu. Rev. Condens. Matter Phys. 7, 119 (2016).
  70. 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).
  71. S. H. Simon, Topological Quantum (Oxford University Press, Oxford, 2023).
  72. F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  73. W. Ji and X.-G. Wen, Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res. 2, 033417 (2020).
  74. Z. Jia, Generalized cluster states from Hopf algebras: Non-invertible symmetry and Hopf tensor network representation, J. High Energy Phys. 09 (2024) 147.
  75. Z. Jia, Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory, J. High Energy Phys. 03 (2026) 154.
  76. O. Buerschaper, J. M. Mombelli, M. Christandl, and M. Aguado, A hierarchy of topological tensor network states, J. Math. Phys. (N.Y.) 54, 012201 (2013).
  77. K. Inamura, On lattice models of gapped phases with fusion category symmetries, J. High Energy Phys. 03 (2022) 036.
  78. D.-C. Lu, A. Chatterjee, and N. Tantivasadakarn, Generalized Kramers-Wannier self-duality in Hopf-Ising models, arXiv:2602.10183.
  79. L. Byles, E. Forbes, and J. K. Pachos, Demonstration of magic state power of D(S3) anyons with two qudits, arXiv:2408.03377.
  80. C. F. B. Lo, A. Lyons, R. Verresen, A. Vishwanath, and N. Tantivasadakarn, Universal quantum computation with the S3 quantum double: A pedagogical exposition, arXiv:2502.14974.
  81. P. Hrmo, B. Wilhelm, L. Gerster, M. W. van Mourik, M. Huber, R. Blatt, P. Schindler, T. Monz, and M. Ringbauer, Native qudit entanglement in a trapped ion quantum processor, Nat. Commun. 14, 2242 (2023).
  82. C. L. Edmunds, E. Rico, I. Arrazola, G. K. Brennen, M. Meth, R. Blatt, and M. Ringbauer, Symmetry-protected topological Haldane phase on a qudit quantum processor, PRX Quantum 6, 020349 (2025).
  83. A. Mariani, S. Pradhan, and E. Ercolessi, Hamiltonians and gauge-invariant Hilbert space for lattice Yang-Mills-like theories with finite gauge group, Phys. Rev. D 107, 114513 (2023).

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Phys. Rev. Lett. 136, 216601– Published 27 May, 2026
  • Received 10 October 2025
  • Revised 13 February 2026
  • Accepted 24 April 2026
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