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Noninvertible Bosonic Chiral Symmetry on the Lattice

Lukasz Fidkowski1, Cenke Xu2, and Carolyn Zhang3

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Phys. Rev. X 16, 041036 – Published 9 November, 2026

DOI: https://doi.org/10.1103/hl1d-99hv

Abstract

In this work, we realize the 3+1D noninvertible ZN chiral symmetry generator as an operator in a many-body lattice Hilbert space. A crucial ingredient in our construction is the use of infinite-dimensional U(1) rotor site Hilbert spaces. Specifically, our Hilbert space is that of a U(1) lattice gauge theory coupled to a charge 1 scalar in the Villain formulation, which allows for direct access to monopoles and for a simple definition of a magnetic ZN one-form symmetry Zm(1), at the lattice Hamiltonian level. We construct the generator of the ZN chiral symmetry as a locality-preserving unitary operator not in the entire Hilbert space but only in the subspace of Zm(1)-invariant states. Using a lattice-level duality based on gauging Zm(1), we find a dual description of this subspace, as the subspace of a charge 1/N gauge theory invariant under an electric one-form symmetry Ze(1). We show that, in this dual formulation, the chiral symmetry generator extends unitarily to the entire Hilbert space but has a mixed anomaly with the Ze(1) symmetry.

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Popular Summary

Realizing noninvertible symmetries as microscopic lattice operators beyond one spatial dimension remains challenging because standard no-go theorems constrain chiral symmetry operations on discrete Hilbert spaces. We bypassed these constraints by constructing a continuous chiral symmetry operator in a three-dimensional bosonic lattice model composed of infinite-dimensional quantum rotors.

We demonstrated that gauging a continuous charge symmetry leaves a noninvertible remnant of chiral symmetry that acts reversibly only within a restricted state subspace. We found that this remnant transformation obeys fusion algebra characteristic of higher-dimensional analogs of Kramers-Wannier duality. Our microscopic construction bridges field-theoretic topological defects with lattice Hamiltonians, opening a pathway to design and simulate phases governed by noninvertible categorical symmetries.

Article Text

References (64)

  1. Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett, Generalized global symmetries, J. High Energy Phys. 02 (2015) 172.
  2. Anton Kapustin and Ryan Thorngren, Higher symmetry and gapped phases of gauge theories, Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift, (Springer, 2017), pp. 177–202.
  3. Yuji Tachikawa, On gauging finite subgroups, SciPost Phys. 8, 015 (2020).
  4. Clay Córdova, Tudor T. Dumitrescu, and Kenneth Intriligator, Exploring 2-group global symmetries, J. High Energy Phys. 02 (2019) 184.
  5. Francesco Benini, Clay Córdova, and Po-Shen Hsin, On 2-group global symmetries and their anomalies, J. High Energy Phys. 03 (2019) 118.
  6. Lakshya Bhardwaj and Yuji Tachikawa, On finite symmetries and their gauging in two dimensions, J. High Energy Phys. 03 (2018) 189.
  7. Chi-Ming Chang, Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, and Xi Yin, Topological defect lines and renormalization group flows in two dimensions, J. High Energy Phys. 01 (2019) 026.
  8. Justin Kaidi, Kantaro Ohmori, and Yunqin Zheng, Kramers-Wannier-like duality defects in (3+1)d gauge theories, Phys. Rev. Lett. 128, 111601 (2022).
  9. Yichul Choi, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, and Shu-Heng Shao, Noninvertible duality defects in 3+1 dimensions, Phys. Rev. D 105, 125016 (2022).
  10. Ryan Thorngren and Yifan Wang, Fusion category symmetry. Part I. Anomaly in-flow and gapped phases, J. High Energy Phys. 04 (2024) 132.
  11. Ryan Thorngren and Yifan Wang, Fusion category symmetry. Part II. Categoriosities at c=1 and beyond, J. High Energy Phys. 07 (2024) 051.
  12. Yichul Choi, Ho Tat Lam, and Shu-Heng Shao, Noninvertible global symmetries in the standard model, Phys. Rev. Lett. 129, 161601 (2022).
  13. Clay Córdova and Kantaro Ohmori, Noninvertible chiral symmetry and exponential hierarchies, Phys. Rev. X 13, 011034 (2023).
  14. Shu-Heng Shao, What’s done cannot be undone: Tasi lectures on non-invertible symmetries, arXiv:2308.00747.
  15. Stephen L. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev. 177, 2426 (1969).
  16. John S. Bell and Roman Jackiw, A pcac puzzle: π0→γγ in the σ-model, Il Nuovo Cimento A (1965-1970) 60, 47 (1969).

  17. We work with a branched simplicial decomposition of a spatial manifold M3, but our results also apply to the ordinary cubic lattice. The only structure we need is that of higher cup products.

  18. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (I). Proof by homotopy theory, Nucl. Phys. B185, 20 (1981).
  19. H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice: (II). Intuitive topological proof, Nucl. Phys. B193, 173 (1981).
  20. Lukasz Fidkowski and Cenke Xu, A no-go result for implementing chiral symmetries by locality-preserving unitaries in a three-dimensional Hamiltonian lattice model of fermions, Phys. Rev. Lett. 131, 196601 (2023).
  21. Lei Gioia and Ryan Thorngren, Exact chiral symmetries of 3+1d Hamiltonian lattice fermions, Phys. Rev. Lett. 136, 061601 (2026).
  22. Arkya Chatterjee, Salvatore D. Pace, and Shu-Heng Shao, Quantized axial charge of staggered fermions and the chiral anomaly, Phys. Rev. Lett. 134, 021601 (2025).
  23. Salvatore D Pace, Arkya Chatterjee, and Shu-Heng Shao, Lattice T-duality from non-invertible symmetries in quantum spin chains, SciPost Phys. 18, 121 (2025).

  24. See also Ref. [25] for a similar result in 1+1D.

  25. Meng Cheng and Nathan Seiberg, Lieb-Schultz-Mattis, Luttinger, and’t Hooft-anomaly matching in lattice systems, SciPost Phys. 15, 051 (2023).
  26. T. Senthil and Michael Levin, Integer quantum Hall effect for bosons, Phys. Rev. Lett. 110, 046801 (2013).

  27. Another useful analogy is to a 1+1-dimensional Dirac fermion, where a 2π axial rotation acts as a translation that pumps charge (the so-called Thouless pump [28, 29]), i.e., a zero-dimensional SPT.

  28. D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983).
  29. Jeffrey Goldstone and Frank Wilczek, Fractional quantum numbers on solitons, Phys. Rev. Lett. 47, 986 (1981).
  30. Michael DeMarco and Xiao-Gang Wen, A commuting projector model with a non-zero quantized Hall conductance, arXiv:2102.13057.
  31. Anton Kapustin and Lukasz Fidkowski, Local commuting projector hamiltonians and the quantum Hall effect, Commun. Math. Phys. 373, 763 (2019).
  32. Carolyn Zhang, Michael Levin, and Sven Bachmann, Vanishing Hall conductance for commuting Hamiltonians, Phys. Rev. B 105, L081103 (2022).

  33. The compactness of ϕ, i.e., the invariance of the wave function under integer shifts of ϕ, follows from the Villain condition and gauge invariance.

  34. Yu-An Chen, Tyler D. Ellison, and Nathanan Tantivasadakarn, Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions, Phys. Rev. Res. 3, 013056 (2021).
  35. Yu-An Chen and Sri Tata, Higher cup products on hypercubic lattices: Application to lattice models of topological phases, J. Math. Phys. (N.Y.) 64, 091902 (2023).
  36. Theodore Jacobson and Tin Sulejmanpasic, Modified villain formulation of Abelian Chern-Simons theory, Phys. Rev. D 107, 125017 (2023).
  37. Michael DeMarco and Xiao-Gang Wen, Compact uk(1) Chern-Simons theory as a local bosonic lattice model with exact discrete 1-symmetries, Phys. Rev. Lett. 126, 021603 (2021).
  38. David Aasen, Roger SK Mong, and Paul Fendley, Topological defects on the lattice: I. The Ising model, J. Phys. A 49, 354001 (2016).
  39. Nathan Seiberg and Shu-Heng Shao, Majorana chain and Ising model-(non-invertible) translations, anomalies, and emanant symmetries, SciPost Phys. 16, 064 (2024).
  40. Nathan Seiberg, Sahand Seifnashri, and Shu-Heng Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space, SciPost Phys. 16, 154 (2024).
  41. Anuj Apte, Clay Córdova, and Ho Tat Lam, Obstructions to gapped phases from noninvertible symmetries, Phys. Rev. B 108, 045134 (2023).
  42. Po-Shen Hsin, Ho Tat Lam, and Nathan Seiberg, Comments on one-form global symmetries and their gauging in 3d and 4d, SciPost Phys. 6, 039 (2019).
  43. Pranay Gorantla, Shu-Heng Shao, and Nathanan Tantivasadakarn, Tensor networks for non-invertible symmetries in 3+1d and beyond, Phys. Rev. X 15, 041006 (2025).
  44. Clay Córdova, Po-Shen Hsin, and Carolyn Zhang, Anomalies of non-invertible symmetries in (3+1) d, SciPost Phys. 17, 131 (2024).
  45. Lokman Tsui and Xiao-Gang Wen, Lattice models that realize zn−1 symmetry-protected topological states for even n, Phys. Rev. B 101, 035101 (2020).
  46. D. Gross, V. Nesme, H. Vogts, and R. F. Werner, Index theory of one dimensional quantum walks and cellular automata, Commun. Math. Phys. 310, 419 (2012).
  47. Zongping Gong, Christoph Sünderhauf, Norbert Schuch, and J. Ignacio Cirac, Classification of matrix-product unitaries with symmetries, Phys. Rev. Lett. 124, 100402 (2020).
  48. Carolyn Zhang, Topological invariants for symmetry-protected topological phase entanglers, Phys. Rev. B 107, 235104 (2023).
  49. Meng Sun, Bowen Yang, Zongyuan Wang, Nathanan Tantivasadakarn, and Yu-An Chen, Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras, PRX Quantum 7, 010362 (2026).
  50. Jeongwan Haah, Lukasz Fidkowski, and Matthew Hastings, Nontrivial quantum cellular automata in higher dimensions, Commun. Math. Phys. 398, 11 (2022).
  51. Ryan Thorngren, John Preskill, and Lukasz Fidkowski, Chiral lattice gauge theories from symmetry disentanglers, arXiv:2601.04304.
  52. Cenke Xu and T. Senthil, Wave functions of bosonic symmetry protected topological phases, Phys. Rev. B 87, 174412 (2013).

  53. The argument is as follows. U1/N(M3), viewed as an operator on He, maps membrane operators that create Ze(1) magnetic flux at their boundaries to themselves multiplied by Wilson loops at their boundaries. But in Hm, the magnetic membrane operators become localized at the boundary (namely, they tunnel a magnetic monopole along the boundary), while the Wilson loops become membrane operators. An operator that maps a loop to a membrane cannot be locality preserving on the full Hilbert space.

  54. Dominic V. Else and Chetan Nayak, Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge, Phys. Rev. B 90, 235137 (2014).
  55. Luis Alvarez-Gaumé and Edward Witten, Gravitational anomalies, Nucl. Phys. B234, 269 (1984).

  56. This follows from the fact that exp(2πiQA(B))|Ψ0⟩=exp(2πiQA(B¯))|Ψ0⟩, where QA(B¯) is the axial charge in the complement of B.

  57. T. Senthil and Michael Levin, Integer quantum Hall effect for bosons, Phys. Rev. Lett. 110, 046801 (2013).

  58. Indeed, if the theory we were dealing with was instead a single Dirac fermion, then it is clear that two conjugate copies of it could be realized in a microscopic Hilbert space with the diagonal U(1)V and U(1)A both acting on site. This is because the field content of such a doubled theory would have four left-handed Weyl fermions, with U(1)V charges +1, −1, −1, and +1 and U(1)A charges +1, +1, −1, and −1. These can be paired up into Dirac fermions (1 with 3 and 2 with 4) such that both symmetries act in a vectorlike way.

  59. Michael Levin, Protected edge modes without symmetry, Phys. Rev. X 3, 021009 (2013).
  60. Yichul Choi, Ho Tat Lam, and Shu-Heng Shao, Noninvertible time-reversal symmetry, Phys. Rev. Lett. 130, 131602 (2023).
  61. Zhiyao Lu, Sahand Seifnashri, and Shu-Heng Shao, Lattice chiral symmetry from bosons in 3+1d, Phys. Rev. D 114, 034518 (2026).
  62. Michael Levin and Zheng-Cheng Gu, Braiding statistics approach to symmetry-protected topological phases, Phys. Rev. B 86, 115109 (2012).
  63. Yitao Feng, Ryohei Kobayashi, Yu-An Chen, and Shinsei Ryu, Higher-form anomalies on lattices, Phys. Rev. Lett. 136, 046504 (2026).
  64. Michael Levin and Xiao-Gang Wen, Quantum ether: Photons and electrons from a rotor model, Phys. Rev. B 73, 035122 (2006).

Outline

Information

Phys. Rev. X 16, 041036– Published 9 November, 2026
  • Received 18 November 2025
  • Revised 1 June 2026
  • Accepted 8 September 2026
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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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