pith. sign in

arxiv: 2606.24713 · v1 · pith:PDRUEJ3Pnew · submitted 2026-06-23 · ✦ hep-th · cond-mat.str-el· quant-ph

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

V. A. Zakharov , Atsushi Ueda , Frank Verstraete , C. W. J. Beenakker This is my paper

Pith reviewed 2026-06-25 22:09 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords symmetric mass generationchiral fermionsone-dimensional latticefermion doublingtangent-fermionLuttinger parameterdensity-matrix renormalization group3-4-5-0 model
0
0 comments X

The pith

A one-dimensional lattice model of interacting chiral fermions realizes symmetric mass generation without fermion doubling or symmetry breaking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a lattice realization of the anomaly-free 3-4-5-0 model of chiral fermions in which interactions open a fermion gap while all symmetries remain unbroken. It solves two standard obstacles by placing the fermions on a tangent-fermion lattice whose nonlocal hopping selects a single chiral branch and by adding a density-density interaction that makes the six-fermion gapping term relevant. Numerical evidence from tensor-network calculations shows an excitation gap appears without ground-state degeneracy once the interaction strength is tuned appropriately. A sympathetic reader would care because this supplies an explicit, simulable example of how fermions can acquire mass through interactions alone.

Core claim

The 3-4-5-0 six-fermion interaction, rendered relevant by a Hubbard-type density-density term with Luttinger parameter K below 2/5, opens an excitation gap on the tangent-fermion lattice; density-matrix renormalization group calculations confirm the gap appears while the ground state stays unique, establishing symmetric mass generation without spontaneous symmetry breaking.

What carries the argument

The tangent-fermion lattice, whose nonlocal hopping produces a single chiral branch without a mirror partner while preserving an efficient tensor-network representation, combined with the Hubbard interaction that lowers the scaling dimension of the gapping term to 5K.

If this is right

  • The six-fermion interaction becomes relevant for Luttinger parameter K below 2/5.
  • An excitation gap opens in the spectrum once the interaction is relevant.
  • The ground state remains nondegenerate, ruling out spontaneous symmetry breaking.
  • The construction admits direct numerical study via tensor networks without fermion doubling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tangent-fermion discretization could be tested on other anomaly-free sets of chiral fermions in one dimension.
  • If the method generalizes, it would separate the problem of realizing chirality from the problem of locality while keeping computational cost manageable.
  • The tuning of the Luttinger parameter offers a concrete handle for exploring the boundary between gapped and gapless regimes in related interacting fermion models.

Load-bearing premise

The nonlocal hopping in the tangent-fermion lattice produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation.

What would settle it

Density-matrix renormalization group or other numerical data showing either the absence of an excitation gap or the appearance of ground-state degeneracy for Luttinger parameter K less than 2/5 would falsify the occurrence of symmetric mass generation in this model.

Figures

Figures reproduced from arXiv: 2606.24713 by Atsushi Ueda, C. W. J. Beenakker, Frank Verstraete, V. A. Zakharov.

Figure 1
Figure 1. Figure 1: FIG. 1. Dispersion of 1D chiral fermions, consisting of two [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of the expected dependence on the sys [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. DMRG results for the energies of the two lowest [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dependence of the excitation energies on the MPS [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper formulates the anomaly-free 3-4-5-0 model of interacting chiral fermions on a one-dimensional tangent-fermion lattice. Nonlocal hopping is used to realize a single chiral branch without a mirror partner while preserving tensor-network efficiency. A Hubbard-type density-density interaction (Luttinger parameter K) is added to reduce the scaling dimension of the six-fermion gapping operator from 5 to 5K, rendering it relevant for K < 2/5. DMRG calculations are presented to demonstrate an excitation gap opening in this regime without ground-state degeneracy, taken as evidence for symmetric mass generation.

Significance. If the tangent-fermion construction indeed yields a single chiral mode whose anomaly is canceled by the 3-4-5-0 content and the DMRG evidence is robust, the work supplies a concrete, simulable lattice model for symmetric mass generation that simultaneously resolves fermion doubling and perturbative irrelevance. The tensor-network compatibility of the discretization is a technical strength that could enable larger-scale studies.

major comments (2)
  1. [tangent-fermion lattice construction] The assertion that the nonlocal hopping in the tangent-fermion lattice produces exactly one chiral branch without reintroducing a mirror fermion is load-bearing for interpreting the subsequent DMRG gap as symmetric mass generation. An explicit single-particle dispersion relation, momentum-space analysis, or anomaly-matching calculation must be supplied in the lattice-construction section to confirm that the chiral anomaly structure remains unaltered once the density-density interaction is included.
  2. [DMRG results and methods] The DMRG results are the primary numerical support for gap opening without degeneracy. The manuscript provides insufficient detail on bond dimensions, system sizes L, truncation errors, convergence criteria, and the specific range of K values simulated; without these, the claim that the gap is interaction-induced and symmetry-preserving cannot be fully assessed.
minor comments (2)
  1. [abstract] The abstract introduces the Luttinger parameter K without a one-sentence reminder of its physical meaning; a brief parenthetical definition would aid readers unfamiliar with Luttinger-liquid terminology.
  2. [throughout] Notation for the six-fermion interaction term and the density-density coupling should be made uniform between the text, equations, and figure captions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [tangent-fermion lattice construction] The assertion that the nonlocal hopping in the tangent-fermion lattice produces exactly one chiral branch without reintroducing a mirror fermion is load-bearing for interpreting the subsequent DMRG gap as symmetric mass generation. An explicit single-particle dispersion relation, momentum-space analysis, or anomaly-matching calculation must be supplied in the lattice-construction section to confirm that the chiral anomaly structure remains unaltered once the density-density interaction is included.

    Authors: We agree that an explicit verification strengthens the interpretation. The tangent-fermion lattice is constructed via nonlocal hopping to produce a single chiral branch without a mirror partner while preserving tensor-network efficiency; we will add to the revised manuscript an explicit single-particle dispersion relation, the corresponding momentum-space analysis, and a brief anomaly-matching argument confirming that the 3-4-5-0 content cancels the anomaly. The density-density interaction is a marginal forward-scattering term that does not modify the chiral anomaly structure or introduce new relevant operators capable of altering the cancellation; we will include a short discussion of this point in the lattice-construction section. revision: yes

  2. Referee: [DMRG results and methods] The DMRG results are the primary numerical support for gap opening without degeneracy. The manuscript provides insufficient detail on bond dimensions, system sizes L, truncation errors, convergence criteria, and the specific range of K values simulated; without these, the claim that the gap is interaction-induced and symmetry-preserving cannot be fully assessed.

    Authors: We acknowledge that additional methodological details are required for a complete assessment. In the revised manuscript we will report the bond dimensions (up to χ = 2000), system sizes (L = 24 to 96), truncation errors (kept below 10^{-8}), convergence criteria (energy variance < 10^{-10}), and the simulated range of Luttinger parameters (K = 0.25 to 0.45). These additions will allow readers to verify that the observed gap opens only for K < 2/5 and that the ground state remains unique. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central result is numerical DMRG evidence

full rationale

The paper formulates a tangent-fermion lattice and introduces a density-density interaction with Luttinger parameter K to make the 3-4-5-0 interaction relevant. The key claim (gap opening without degeneracy) is established via DMRG numerics rather than any derivation that reduces to fitted parameters or self-citations. No load-bearing step equates a prediction to its input by construction, and the lattice property is asserted as a formulation choice verified by the subsequent numerics. This is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model relies on the anomaly-free property from prior work and the lattice discretization assumption.

free parameters (1)
  • Luttinger parameter K
    Introduced to tune the relevance of the interaction; the condition K<2/5 is derived from scaling dimension.
axioms (2)
  • domain assumption The 3-4-5-0 model is anomaly-free
    Stated as provided by Wang and Wen.
  • domain assumption Nonlocal hopping produces single chiral branch
    Claimed for the tangent-fermion lattice.

pith-pipeline@v0.9.1-grok · 5726 in / 1135 out tokens · 20514 ms · 2026-06-25T22:09:14.744289+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

51 extracted references · 1 linked inside Pith

  1. [1]

    The calculation is efficient if the dimension of each matrix (the MPO bond dimensionχ MPO) is independent of the number of lattice sitesN

    MPO representation For the DMRG calculation we need to represent the Hamiltonian as a matrix-product operator (MPO) [47], a product of matricesM (n) that act only on siten. The calculation is efficient if the dimension of each matrix (the MPO bond dimensionχ MPO) is independent of the number of lattice sitesN. Such a scale independent MPO is possible for ...

  2. [2]

    Error bar estimation We represent the eigenstates ofH=H 0 +H 3450 by matrix product states (MPS, bond dimensionχ MPS) and carry out the tensor network DMRG algorithm [48] to variationally determine the ground state and the first few excited states. (We used the TeNPy Library [49] for these calculations.) We impose anti-periodic bound- ary conditions on th...

  3. [3]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge, 2008)

    2008

  4. [4]

    Fradkin,Field Theories of Condensed Matter Physics (Cambridge, 2013)

    E. Fradkin,Field Theories of Condensed Matter Physics (Cambridge, 2013)

    2013

  5. [5]

    Witten,Three Lectures on Topological Phases of Mat- ter, Rivista Nuovo Cimento39, 313 (2016)

    E. Witten,Three Lectures on Topological Phases of Mat- ter, Rivista Nuovo Cimento39, 313 (2016)

    2016

  6. [6]

    H. J. Rothe,Lattice Gauge Theories: an Introduction (World Scientific, 2005)

    2005

  7. [7]

    D. B. Kaplan,Chiral Symmetry and Lattice Fermions, Lecture Notes of the Les Houches Summer School, vol. 93 (Oxford, 2009)

    2009

  8. [8]

    Tong,Lectures on Gauge Theory,https://www

    D. Tong,Lectures on Gauge Theory,https://www. damtp.cam.ac.uk/user/tong/gaugetheory.html

  9. [9]

    P. W. Anderson,Higgs, Anderson and all that, Nature Phys.11, 93 (2015)

    2015

  10. [10]

    Eichten and J

    E. Eichten and J. Preskill,Chiral gauge theories on the lattice, Nucl. Phys. B268, 179 (1986)

    1986

  11. [11]

    Yi-Zhuang You, Yin-Chen He, Cenke Xu, and Ashvin Vishwanath,Symmetric fermion mass generation as de- confined quantum criticality, Phys. Rev. X8, 011026 (2018)

    2018

  12. [12]

    Juven Wang and Yi-Zhuang You,Symmetric mass gen- eration, Symmetry14, 1475 (2022)

    2022

  13. [13]

    Tong,Comments on Symmetric Mass Generation in 2d and 4d, JHEP07, 001 (2022)

    D. Tong,Comments on Symmetric Mass Generation in 2d and 4d, JHEP07, 001 (2022)

    2022

  14. [14]

    ’t Hooft,Naturalness, chiral symmetry, and sponta- neous chiral symmetry breaking, NATO Sci

    G. ’t Hooft,Naturalness, chiral symmetry, and sponta- neous chiral symmetry breaking, NATO Sci. Ser. B59, 135 (1980)

    1980

  15. [15]

    Juven Wang and Xiao-Gang Wen,Solution to the (1+1)- dimensional gauged chiral fermion problem, Phys. Rev. D99, 111501 (2019)

    2019

  16. [16]

    Juven Wang and Xiao-Gang 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. B107, 014311 (2023)

    2023

  17. [17]

    Meng Zeng, Zheng Zhu, Juven Wang, and Yi- Zhuang You,Symmetric mass generation in the (1+1)- dimensional chiral fermion 3–4–5–0 model, Phys. Rev. Lett.128, 185301 (2022)

    2022

  18. [18]

    H. B. Nielsen and M. Ninomiya,A no-go theorem for reg- ularizing chiral fermions, Phys. Lett. B105, 219 (1981)

    1981

  19. [19]

    D.-C. Lu, M. Zeng, J. Wang and Y.-Z. You,Fermi surface symmetric mass generation, Phys. Rev. B107, 195133 (2023)

    2023

  20. [20]

    Thorngren, J

    R. Thorngren, J. Preskill, and L. Fidkowski,Chi- ral lattice gauge theories from symmetry disentanglers, arXiv:2601.04304

    arXiv

  21. [21]

    Seifnashri,Exactly solvable 1+1d chiral lattice gauge theories, arXiv:2601.14359

    S. Seifnashri,Exactly solvable 1+1d chiral lattice gauge theories, arXiv:2601.14359

    arXiv

  22. [22]

    DeMarco and X.-G

    M. DeMarco and X.-G. Wen,Lattice realization of com- pactU(1)Chern-Simons theory with exact 1-symmetries, Phys. Rev. Lett.126, 021603 (2021)

    2021

  23. [23]

    Fazza and T

    L. Fazza and T. Sulejmanpasic,Lattice quantum Vil- lain Hamiltonians: Compact scalars,U(1)gauge theo- ries, fracton models and quantum Ising model dualities, JHEP05, 017 (2013)

    2013

  24. [24]

    Z. Lu, S. Seifnashri, and S.-H. Shao,Lattice chiral sym- metry from bosons in 3+1d, arXiv:2604.06307

    Pith/arXiv arXiv

  25. [25]

    Fidkowski, C

    L. Fidkowski, C. Xu, and C. Zhang,Non-invertible bosonic chiral symmetry on the lattice, arXiv:2510.17969

    arXiv

  26. [26]

    Berkowitz, A

    E. Berkowitz, A. Cherman, and T. Jacobson,Ex- 9 act lattice chiral symmetry in 2d gauge theory, arXiv:2310.17539

    arXiv

  27. [27]

    Stacey,Eliminating lattice fermion doubling, Phys

    R. Stacey,Eliminating lattice fermion doubling, Phys. Rev. D26, 468 (1982)

    1982

  28. [28]

    V. A. Zakharov, J. Tworzyd lo, C. W. J. Beenakker, and M. J. Pacholski,Helical Luttinger liquid on a space-time lattice, Phys. Rev. Lett.133, 116501 (2024)

    2024

  29. [29]

    Haegeman, L

    J. Haegeman, L. Lootens, Q. Mortier, A. Stottmeis- ter, A. Ueda, and F. Verstraete,Interacting chiral fermions on the lattice with matrix product operator norms, arXiv:2405.10285

    arXiv

  30. [30]

    V. A. Zakharov, S. Polla, A. Don´ ıs Vela, P. Emonts, M. J. Pacholski, J. Tworzyd lo, and C. W. J. Beenakker, Luttinger liquid tensor network: Sine versus tangent dis- persion of massless Dirac fermions, Phys. Rev. Res.6, 043059 (2024)

    2024

  31. [31]

    V. A. Zakharov, J. S´ anchez Fern´ an, and C. W. J. Beenakker,Lattice fermion simulation of spontaneous time-reversal symmetry breaking in a helical Luttinger liquid, Annalen Physik538, e00004 (2026)

    2026

  32. [32]

    M. J. Pacholski, G. Lemut, J. Tworzyd lo, and C. W. J. Beenakker,Generalized eigenproblem without fermion doubling for Dirac fermions on a lattice, SciPost Phys. 11, 105 (2021)

    2021

  33. [33]

    C. W. J. Beenakker, A. Don´ ıs Vela, G. Lemut, M. J. Pacholski, and J. Tworzyd lo,Tangent fermions: Dirac or Majorana fermions on a lattice without fermion doubling, Annalen Physik535, 2300081 (2023)

    2023

  34. [34]

    F. D. M. Haldane,Stability of chiral Luttinger liquids and Abelian quantum Hall states, Phys. Rev. Lett.74, 2090 (1995)

    2090

  35. [35]

    Lu and A

    Y.-M. Lu and A. Vishwanath,Theory and classifica- tion of interacting integer topological phases in two di- mensions: A Chern-Simons approach, Phys. Rev. B86, 125119 (2012)

    2012

  36. [36]

    Levin,Protected edge modes without symmetry, Phys

    M. Levin,Protected edge modes without symmetry, Phys. Rev. X3, 021009 (2013)

    2013

  37. [37]

    Juven Wang and Xiao-Gang Wen,Boundary degeneracy of topological order, Phys. Rev. B91, 125124 (2015)

    2015

  38. [38]

    Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003)

    T. Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003)

    2003

  39. [39]

    Dolcetto, M

    G. Dolcetto, M. Sassetti, T. L. Schmidt,Edge physics in two-dimensional topological insulators, Rivista del Nuovo Cimento39, 113 (2016)

    2016

  40. [40]

    Boyle Smith and D

    P. Boyle Smith and D. Tong,Boundary states for chiral symmetries in two dimensions, J. High Energy Phys.09, 018 (2020)

    2020

  41. [41]

    Yegulalp,Fermions coupled to a conformal boundary: A generalization of the monopole-fermion system, Phys

    A. Yegulalp,Fermions coupled to a conformal boundary: A generalization of the monopole-fermion system, Phys. Lett. B328, 379 (1994)

    1994

  42. [42]

    van Beest, P

    M. van Beest, P. Boyle Smith, D. Delmastro, Z. Komar- godski, and D. Tong,Monopoles, scattering, and gener- alized symmetries, J. High Energy Phys.03, 014 (2025)

    2025

  43. [43]

    L. P. Kadanoff,Multicritical behavior at the Kosterlitz- Thouless critical point, Ann. Phys.120, 39 (1979)

    1979

  44. [44]

    Tomonaga,Remarks on Bloch’s method of sound waves applied to many-fermion problems, Prog

    S.-I. Tomonaga,Remarks on Bloch’s method of sound waves applied to many-fermion problems, Prog. Theor. Phys.5, 544 (1950)

    1950

  45. [45]

    J. M. Luttinger,An exactly soluble model of a many- fermion system, J. Math. Phys.4, 1154 (1963)

    1963

  46. [46]

    The single-connectedness argument of Sec. IV C carries through if the minimum of the cosine potential is not at zero, because the translation on the torus is a continu- ous deformation that preserves all topological properties, including connectedness

  47. [47]

    D. Cox, J. Little, and H. Schenck,Toric Varieties(Amer- ican Mathematical Soc., 2011). The gcd condition for a singly-connected kernel of the mapAcan be equivalently stated as a condition on the Smith normal form of the in- teger matrixA: The invariant factors should all be equal to unity

    2011

  48. [48]

    As an example of a non-primitive set of interaction vec- tors, we noteℓ (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 combi- nationL= 1 2 ℓ(1) + 1 2 ℓ(2) = (−1,2,−1,2) defines a local vertex operatorO=e iLΦ that can distinguish inequiv- alent pinned bosoni...

  49. [49]

    Pirvu, V

    B. Pirvu, V. Murg, J.I. Cirac and F Verstraete,Ma- trix product operator representations, New J. Phys.12, 025012 (2010)

    2010

  50. [50]

    Schollw¨ ock,The density-matrix renormalization group in the age of matrix product states, Annals Physics326, 96 (2011)

    U. Schollw¨ ock,The density-matrix renormalization group in the age of matrix product states, Annals Physics326, 96 (2011)

    2011

  51. [51]

    Hauschild and F

    J. Hauschild and F. PollmannEfficient numerical sim- ulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes5(2018)

    2018