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Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling

V. A. Zakharov1,*,†, Atsushi Ueda2,*, Frank Verstraete2,3, and C. W. J. Beenakker1

  • 1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
  • 2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
  • 3Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

  • *These authors contributed equally to this work.
  • †Contact author: zakharov@lorentz.leidenuniv.nl
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Phys. Rev. Research 8, 043009 – Published 5 October, 2026

DOI: https://doi.org/10.1103/2v9h-25qs

Abstract

Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3–4–5–0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter K) that reduces the scaling dimension of the 3–4–5–0 interaction from 5 to 5K, making it relevant for K<2/5. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.

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References (55)

  1. G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2008).
  2. E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, Cambridge, 2013).
  3. E. Witten, Three lectures on topological phases of matter, Riv. Nuovo Cimento 39, 313 (2016).
  4. H. J. Rothe, Lattice Gauge Theories: An Introduction (World Scientific, Singapore, 2005).
  5. D. B. Kaplan, Chiral Symmetry and Lattice Fermions, Lecture Notes of the Les Houches Summer School (Oxford University Press, Oxford, 2009), Vol. 93.
  6. D. Tong, Lectures on gauge theory, 2018, https://www.damtp.cam.ac.uk/user/tong/gaugetheory.html.
  7. P. W. Anderson, Higgs, Anderson and all that, Nat. Phys. 11, 93 (2015).
  8. E. Eichten and J. Preskill, Chiral gauge theories on the lattice, Nucl. Phys. B 268, 179 (1986).
  9. X.-G. Wen, A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model, Chin. Phys. Lett. 30, 111101 (2013).
  10. Y.-Z. You, Y.-C. He, C. Xu, and A. Vishwanath, Symmetric fermion mass generation as deconfined quantum criticality, Phys. Rev. X 8, 011026 (2018).
  11. J. Wang and Y.-Z. You, Symmetric mass generation, Symmetry 14, 1475 (2022).
  12. D. Tong, Comments on symmetric mass generation in 2d and 4d, J. High Energy Phys. 07 (2022) 001.
  13. R. Mouland, D. Tong, and B. Zan, Phases of 2d gauge theories and symmetric mass generation, J. High Energy Phys. 04 (2026) 154.
  14. A. Hasenfratz and C. Xu, A guide to symmetric mass generation in lattice-QCD, Annu. Phys. 495, 170730 (2026).
  15. G.’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, NATO Sci. Ser. B 59, 135 (1980).
  16. J. Wang and X.-G. Wen, Solution to the (1 + 1)-dimensional gauged chiral fermion problem, Phys. Rev. D 99, 111501(R) (2019).
  17. J. Wang and X.-G. Wen, Non-perturbative regularization of (1 + 1)-dimensional anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules, Phys. Rev. B 107, 014311 (2023).
  18. M. Zeng, Z. Zhu, J. Wang, and Y.-Z. You, Symmetric mass generation in the (1 + 1)-dimensional chiral fermion 3–4–5–0 model, Phys. Rev. Lett. 128, 185301 (2022).
  19. H. B. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, Phys. Lett. B 105, 219 (1981).
  20. D.-C. Lu, M. Zeng, J. Wang, and Y.-Z. You, Fermi surface symmetric mass generation, Phys. Rev. B 107, 195133 (2023).
  21. R. Thorngren, J. Preskill, and L. Fidkowski, Chiral lattice gauge theories from symmetry disentanglers, arXiv:2601.04304.
  22. S. Seifnashri, Exactly solvable 1+1d chiral lattice gauge theories, arXiv:2601.14359.
  23. M. DeMarco and X.-G. Wen, Lattice realization of compact U(1) Chern-Simons theory with exact 1-symmetries, Phys. Rev. Lett. 126, 021603 (2021).
  24. L. Fazza and T. Sulejmanpasic, Lattice quantum Villain Hamiltonians: Compact scalars, U(1) gauge theories, fracton models and quantum Ising model dualities, J. High Energy Phys. 05 (2023) 017.
  25. Z. Lu, S. Seifnashri, and S.-H. Shao, Lattice chiral symmetry from bosons in 3+1d, Phys. Rev. D 114, 034518 (2026).
  26. L. Fidkowski, C. Xu, and C. Zhang, Non-invertible bosonic chiral symmetry on the lattice, arXiv:2510.17969.
  27. E. Berkowitz, A. Cherman, and T. Jacobson, Exact lattice chiral symmetry in 2d gauge theory, arXiv:2310.17539.
  28. Z. Lu and S.-H. Shao, Fermionic Villain model with exact lattice chiral symmetries, arXiv:2608.02728.
  29. T. Dharanikota and L. Fidkowski, 1 + 1d lattice Dirac fermions from non-onsite vector and axial symmetries, arXiv:2608.02722.
  30. R. Stacey, Eliminating lattice fermion doubling, Phys. Rev. D 26, 468 (1982).
  31. V. A. Zakharov, J. Tworzydło, C. W. J. Beenakker, and M. J. Pacholski, Helical Luttinger liquid on a space-time lattice, Phys. Rev. Lett. 133, 116501 (2024).
  32. J. Haegeman, L. Lootens, Q. Mortier, A. Stottmeister, A. Ueda, and F. Verstraete, Interacting chiral fermions on the lattice with matrix product operator norms, arXiv:2405.10285.
  33. V. A. Zakharov, S. Polla, A. Donís Vela, P. Emonts, M. J. Pacholski, J. Tworzydło, and C. W. J. Beenakker, Luttinger liquid tensor network: Sine versus tangent dispersion of massless Dirac fermions, Phys. Rev. Res. 6, 043059 (2024).
  34. V. A. Zakharov, J. S. Fernán, and C. W. J. Beenakker, Lattice fermion simulation of spontaneous time-reversal symmetry breaking in a helical Luttinger liquid, Ann. Phys. 538, e00004 (2026).
  35. M. J. Pacholski, G. Lemut, J. Tworzydło, and C. W. J. Beenakker, Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice, SciPost Phys. 11, 105 (2021).
  36. C. W. J. Beenakker, A. D. Vela, G. Lemut, M. J. Pacholski, and J. Tworzydło, Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling, Ann. Phys. 535, 2300081 (2023).
  37. F. D. M. Haldane, Stability of chiral Luttinger liquids and Abelian quantum Hall states, Phys. Rev. Lett. 74, 2090 (1995).
  38. Y.-M. Lu and A. Vishwanath, Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach, Phys. Rev. B 86, 125119 (2012).
  39. M. Levin, Protected edge modes without symmetry, Phys. Rev. X 3, 021009 (2013).
  40. J. Wang and X.-G. Wen, Boundary degeneracy of topological order, Phys. Rev. B 91, 125124 (2015).
  41. T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003).
  42. G. Dolcetto, M. Sassetti, and T. L. Schmidt, Edge physics in two-dimensional topological insulators, Riv. Nuovo Cimento 39, 113 (2016).
  43. P. B. Smith and D. Tong, Boundary states for chiral symmetries in two dimensions, J. High Energy Phys. 09 (2020) 018.
  44. A. Yegulalp, Fermions coupled to a conformal boundary: A generalization of the monopole-fermion system, Phys. Lett. B 328, 379 (1994).
  45. M. van Beest, P. B. Smith, D. Delmastro, Z. Komargodski, and D. Tong, Monopoles, scattering, and generalized symmetries, J. High Energy Phys. 03 (2025) 014.
  46. L. P. Kadanoff, Multicritical behavior at the Kosterlitz-Thouless critical point, Ann. Phys. 120, 39 (1979).
  47. S.-I. Tomonaga, Remarks on Bloch’s method of sound waves applied to many-fermion problems, Prog. Theor. Phys. 5, 544 (1950).
  48. J. M. Luttinger, An exactly soluble model of a many-fermion system, J. Math. Phys. 4, 1154 (1963).
  49. The single-connectedness argument of Sec. 4c carries through if the minimum of the cosine potential is not at zero, because the translation on the torus is a continuous deformation that preserves all topological properties, including connectedness.
  50. D. Cox, J. Little, and H. Schenck, Toric Varieties (American Mathematical Soc., Providence, Rhode Island, 2011). The gcd condition for a singly connected kernel of the map A can be equivalently stated as a condition on the Smith normal form of the integer matrix A: The invariant factors should all be equal to unity.
  51. As an example of a nonprimitive set of interaction vectors, we note ℓ(1)=(1,3,−3,−1), ℓ(2)=(−3,1,1,−3). These satisfy the conditions (2.4) and (2.5), but they do not span a primitive lattice: The fractional linear combination L=12ℓ(1)+12ℓ(2)=(−1,2,−1,−2) defines a local vertex operator O=eiLΦ that can distinguish inequivalent pinned bosonic fields: If cos(ℓ(p)Φ)=2πnp, then O=(−1)n1+n2 takes on the value ±1 on the two inequivalent minima of the cosine potential. This is an accidental ground-state degeneracy, which can be removed by including in the Hamiltonian the operator O to favor one of the two minima without breaking U(1) symmetry, hence without compromising SMG.
  52. V. A. Zakharov, A. Ueda, F. Verstraete, and C. Beenakker, Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling, [Computer software], Zenodo, 2026, https://doi.org/10.5281/zenodo.20816064.
  53. B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete, Matrix product operator representations, New J. Phys. 12, 025012 (2010).
  54. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  55. J. Hauschild and F. Pollmann, Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes 5 (2018).

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Phys. Rev. Research 8, 043009– Published 5 October, 2026
  • Received 26 June 2026
  • Accepted 5 September 2026
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