pith. machine review for the scientific record. sign in

arxiv: 2604.02078 · v2 · submitted 2026-04-02 · ✦ hep-lat · cond-mat.str-el· hep-th

Recognition: no theorem link

Taste-splitting mass and edge modes in 3+1 D staggered fermions

Tatsuhiro Misumi, Tatsuya Yamaoka, Tetsuya Onogi

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:59 UTC · model grok-4.3

classification ✦ hep-lat cond-mat.str-elhep-th
keywords staggered fermionsdomain wallparity anomalyflavor symmetryedge modestaste splittinglattice Hamiltonian
0
0 comments X

The pith

Staggered fermions with a one-link mass kink host two-flavor massless Dirac fermions on the domain wall.

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

This paper studies the symmetry structure of local mass terms in the 3+1 dimensional staggered fermion Hamiltonian on the lattice. It identifies a one-link mass term that preserves an enlarged symmetry, including conserved charges from the Onsager algebra. Introducing a kink in this mass term gaps the bulk while localizing two-flavor massless Dirac fermions on the 2+1 dimensional domain wall. The bulk symmetries induce an SU(2) flavor symmetry on the wall modes, which together with reflection symmetry prohibits any symmetric mass term, realizing the parity anomaly of the boundary theory from the ultraviolet lattice theory rather than emerging only at low energies.

Core claim

In the staggered fermion system, a domain wall created by a kink profile in one of the one-link mass terms results in a gapped bulk with two-flavor massless Dirac fermions localized on the wall. The conserved charges from the bulk act as SU(2) flavor generators on the boundary, and imposing both this flavor SU(2) and space reflection symmetry forbids a symmetric mass gap for the wall modes. This demonstrates that the flavor symmetry and parity anomaly of the boundary theory are inherited from the ultraviolet staggered-fermion Hamiltonian.

What carries the argument

The one-link bilinear mass term with a kink profile, whose preserved symmetries include generators that descend to an SU(2) flavor symmetry on the domain wall.

If this is right

  • The boundary theory remains massless under the inherited symmetries.
  • This lattice setup provides a UV origin for the parity anomaly in the 2+1D theory.
  • Other mass terms split tastes but preserve fewer symmetries than the one-link term.
  • The bulk charges generate the full flavor SU(2) acting on the wall modes.

Where Pith is reading between the lines

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

  • Similar kink constructions in staggered fermions could regularize anomalous theories in other dimensions without fine-tuning.
  • The Onsager algebra structure preserved by the mass term may imply exact solvability or integrability for the domain-wall spectrum.
  • This mechanism could extend to gauge-coupled versions to protect chiral edge states in lattice models of topological phases.

Load-bearing premise

The low-energy continuum limit on the domain wall yields exactly two-flavor massless Dirac fermions whose only allowed mass terms are forbidden by the inherited SU(2) and reflection symmetries.

What would settle it

A calculation showing that the two-flavor wall modes can acquire a symmetric mass gap while preserving both the SU(2) flavor symmetry and reflection symmetry would falsify the claim.

read the original abstract

We investigate the symmetry structure of the $3+1$ D staggered fermion Hamiltonian and its implications for anomalies. Since the spin and flavor degrees of freedom of Dirac fermions are distributed over the lattice, in addition to the standard on-site mass term, the staggered fermion system also admits one-, two-, and three-link bilinear terms within a unit cube as local, charge conserving mass terms with different spin and flavor dependence. We identify the spin flavor structures of all those bilinear mass terms and determine the symmetries preserved by each of them. Among them, one of the one-link mass terms preserves a larger residual symmetry associated with conserved charges that generate the Onsager algebra. Motivated by this structure, we consider a kink profile of the one-link mass and analyze the resulting domain-wall system. In the low-energy limit, the $3+1$ D bulk becomes gapped, while two-flavor massless Dirac fermions appear as localized modes on the $2+1$ D domain wall. We show that the bulk conserved charges act on the wall as generators of a flavor $\mathrm{SU}(2)$ symmetry, and that no symmetric mass gap is allowed for the boundary theory when this $\mathrm{SU}(2)$ symmetry and space reflection symmetry are both imposed. This realizes the parity anomaly of the boundary theory and shows that the boundary flavor symmetry and anomaly descend from the ultraviolet staggered-fermion Hamiltonian rather than emerging only in the infrared.

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 / 1 minor

Summary. The manuscript analyzes the symmetry properties of various bilinear mass terms in the 3+1D staggered fermion Hamiltonian, identifying spin-flavor structures for on-site, one-, two-, and three-link terms. It highlights a one-link mass term preserving an enlarged symmetry tied to the Onsager algebra. A kink profile in this term is introduced to form a domain wall, where the 3+1D bulk gaps out while two-flavor massless Dirac fermions localize on the 2+1D wall. Bulk conserved charges are shown to act as SU(2) flavor generators on the wall, and the combination of this SU(2) with space reflection is argued to forbid any symmetric mass gap, realizing the parity anomaly directly from the UV staggered Hamiltonian.

Significance. If the central claims are verified, the work supplies a concrete UV lattice derivation of boundary flavor symmetry and anomaly protection in a staggered-fermion domain-wall setup. This strengthens the link between lattice symmetries and continuum anomalies, with potential utility for understanding edge modes in lattice QCD and condensed-matter models. The explicit construction from the staggered Hamiltonian, without ad-hoc parameters, is a notable strength.

major comments (2)
  1. [Domain-wall construction and boundary effective theory] The central no-symmetric-mass-gap claim for the boundary theory (abstract and domain-wall section) rests on the low-energy effective description being precisely two-flavor massless Dirac fermions whose only allowed mass bilinears are ruled out by the inherited SU(2) and reflection symmetries. The manuscript does not enumerate all local, charge-conserving, symmetry-invariant operators (including O(a) and O(a^2) taste-mixing terms) that could appear in the effective wall Hamiltonian and potentially open a gap while respecting the UV symmetries.
  2. [Symmetry action on wall modes] The statement that bulk conserved charges generate an SU(2) flavor symmetry acting on the wall modes requires explicit algebraic verification of the representation on the localized edge states and confirmation that no additional lattice artifacts modify this action; the current description remains at the level of symmetry classification without the full commutation relations or mode expansion.
minor comments (1)
  1. [Bilinear mass terms] Explicit matrix representations or component expansions for the one-, two-, and three-link mass bilinears would improve clarity when identifying their spin-flavor content and preserved symmetries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to strengthen the presentation of the domain-wall construction and symmetry analysis.

read point-by-point responses

  1. Referee: The central no-symmetric-mass-gap claim for the boundary theory (abstract and domain-wall section) rests on the low-energy effective description being precisely two-flavor massless Dirac fermions whose only allowed mass bilinears are ruled out by the inherited SU(2) and reflection symmetries. The manuscript does not enumerate all local, charge-conserving, symmetry-invariant operators (including O(a) and O(a^2) taste-mixing terms) that could appear in the effective wall Hamiltonian and potentially open a gap while respecting the UV symmetries.

    Authors: We agree that an explicit enumeration of all symmetry-allowed local operators in the effective wall Hamiltonian, including higher-order taste-mixing terms, would make the no-gap argument more rigorous. In the revised manuscript we will add a dedicated subsection classifying all charge-conserving, SU(2)- and reflection-invariant bilinears up to O(a^2). We will show that the only relevant operators are the two-flavor Dirac mass terms already considered, while all taste-mixing and higher-dimension operators either violate the symmetries or are irrelevant for gapping the modes at the scales of interest. This confirms that the protection against a symmetric mass gap is robust. revision: yes

  2. Referee: The statement that bulk conserved charges generate an SU(2) flavor symmetry acting on the wall modes requires explicit algebraic verification of the representation on the localized edge states and confirmation that no additional lattice artifacts modify this action; the current description remains at the level of symmetry classification without the full commutation relations or mode expansion.

    Authors: We acknowledge that the current text presents the symmetry action primarily through classification. In the revision we will supply the explicit algebraic verification: we will derive the commutation relations between the bulk conserved charges (generators of the Onsager algebra) and the creation/annihilation operators for the localized wall modes, obtained from the mode expansion of the domain-wall solutions. This will demonstrate that the wall modes transform in the fundamental representation of SU(2) with no additional lattice artifacts altering the action at leading order in the low-energy expansion. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit UV Hamiltonian derivation is self-contained

full rationale

The paper begins with the explicit 3+1D staggered-fermion Hamiltonian, enumerates all local bilinear mass terms (on-site, one-, two-, and three-link) within a unit cube, determines the symmetries each preserves, selects the one-link term that preserves the largest residual symmetry (Onsager algebra charges), imposes a kink profile, and analyzes the resulting domain-wall spectrum. The boundary SU(2) generators and reflection symmetry are shown to descend directly from the bulk conserved charges, forbidding symmetric mass terms for the two-flavor wall modes. No parameters are fitted to data, no quantity is renamed as a prediction after being input, and no load-bearing result is justified solely by self-citation. The derivation consists of direct symmetry classification and low-energy mode analysis from the given lattice operator, remaining independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard lattice assumptions of locality, charge conservation, and the distribution of spin-flavor degrees of freedom across sites; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Staggered fermions distribute spin and flavor degrees of freedom across lattice sites within a unit cube
    Standard starting point for staggered-fermion Hamiltonians in 3+1D
  • domain assumption Local, charge-conserving mass terms are restricted to on-site and one-, two-, three-link bilinears inside each unit cube
    Follows from the requirement of locality and gauge invariance on the lattice

pith-pipeline@v0.9.0 · 5569 in / 1424 out tokens · 51266 ms · 2026-05-13T20:59:09.109541+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Lattice fermion formulation via Physics-Informed Neural Networks: Ginsparg-Wilson relation and Overlap fermions

    hep-lat 2026-05 unverdicted novelty 7.0

    Physics-informed neural networks construct overlap fermions by optimizing to the Ginsparg-Wilson relation and autonomously discover both the standard and generalized Fujikawa-type versions of the relation.

  2. Lattice fermion formulation via Physics-Informed Neural Networks: Ginsparg-Wilson relation and Overlap fermions

    hep-lat 2026-05 unverdicted novelty 7.0

    Physics-Informed Neural Networks construct lattice Dirac operators satisfying the Ginsparg-Wilson relation, reproducing overlap fermions to high accuracy and discovering a Fujikawa-type generalized relation via algebr...

  3. Lattice chiral symmetry from bosons in 3+1d

    hep-th 2026-04 unverdicted novelty 7.0

    A bosonic lattice model realizes exact chiral symmetry and its anomaly in 3+1d, with the continuum limit a compact boson theory with axion-like coupling.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages · cited by 2 Pith papers · 10 internal anchors

  1. [1]

    Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory,

    H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory,” Nucl. Phys. B185(1981) 20. [Erratum: Nucl.Phys.B 195, 541 (1982)]

    work page 1981

  2. [2]

    No Go Theorem for Regularizing Chiral Fermions,

    H. B. Nielsen and M. Ninomiya, “No Go Theorem for Regularizing Chiral Fermions,” Phys. Lett. B105(1981) 219–223

    work page 1981

  3. [3]

    Quarks and Strings on a Lattice,

    K. G. Wilson, “Quarks and Strings on a Lattice,” in 13th International School of Subnuclear Physics: New Phenomena in Subnuclear Physics. 11, 1975

    work page 1975

  4. [4]

    Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,

    J. B. Kogut and L. Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,” Phys. Rev. D11(1975) 395–408

    work page 1975

  5. [5]

    Lattice Fermions,

    L. Susskind, “Lattice Fermions,” Phys. Rev. D16(1977) 3031–3039

    work page 1977

  6. [6]

    Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation,

    Y.-Y. Li, J. Wang, and Y.-Z. You, “Quantum Many-Body Lattice C-R-T Symmetry: Fractionalization, Anomaly, and Symmetric Mass Generation,”2412.19691

    work page arXiv

  7. [7]

    Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly,

    A. Chatterjee, S. D. Pace, and S.-H. Shao, “Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly,” Phys. Rev. Lett.134(2025), no. 2, 021601,2409.12220

    work page arXiv 2025

  8. [8]

    Symmetries and Anomalies of Hamiltonian Staggered Fermions,

    S. Catterall, A. Pradhan, and A. Samlodia, “Symmetries and Anomalies of Hamiltonian Staggered Fermions,”2501.10862

    work page arXiv

  9. [9]

    Quantized axial charge in the Hamiltonian approach to Wilson fermions,

    T. Yamaoka, “Quantized Axial Charge in the Hamiltonian Approach to Wilson Fermions,” 2504.10263

    work page arXiv

  10. [10]

    Non-singlet conserved charges and anomalies in 3+1 D staggered fermions,

    T. Onogi and T. Yamaoka, “Non-singlet conserved charges and anomalies in 3+1 D staggered fermions,”2509.04906

    work page arXiv

  11. [11]

    Chiral Anomaly of Kogut-Susskind Fermion in (3+1)-dimensional Hamiltonian formalism,

    S. Aoki, Y. Kikukawa, and T. Takemoto, “Chiral anomaly of Kogut-Susskind fermions in the (3+1)-dimensional Hamiltonian formalism,” Phys. Rev. D113(2026), no. 3, 034514, 2511.06198

    work page arXiv 2026

  12. [12]

    Tori, Klein Bottles, and Modulo 8 Parity/Time-reversal Anomalies of 2+1d Staggered Fermions

    N. Seiberg and W. Zhang, “Tori, Klein Bottles, and Modulo 8 Parity/Time-reversal Anomalies of 2+1d Staggered Fermions,”2601.01191

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Lattice gauge theory for the Haldane conjecture and central-branch Wilson fermion,

    T. Misumi and Y. Tanizaki, “Lattice gauge theory for the Haldane conjecture and central-branch Wilson fermion,” PTEP2020(2020), no. 3, 033B03,1910.09604

    work page arXiv 2020

  14. [14]

    Crystal statistics. 1. A Two-dimensional model with an order disorder transition,

    L. Onsager, “Crystal statistics. 1. A Two-dimensional model with an order disorder transition,” Phys. Rev.65(1944) 117–149

    work page 1944

  15. [15]

    Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions

    L. Gioia and R. Thorngren, “Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions,”2503.07708

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [16]

    Parity anomaly from LSM: exact valley symmetries on the lattice,

    S. D. Pace, M. L. Kim, A. Chatterjee, and S.-H. Shao, “Parity anomaly from LSM: exact valley symmetries on the lattice,”2505.04684

    work page arXiv

  17. [17]

    Two soluble models of an antiferromagnetic chain,

    E. H. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Annals Phys.16(1961) 407–466

    work page 1961

  18. [18]

    A Proof of Part of Haldane’s Conjecture on Spin Chains,

    I. Affleck and E. H. Lieb, “A Proof of Part of Haldane’s Conjecture on Spin Chains,” Lett. Math. Phys.12(1986) 57

    work page 1986

  19. [19]

    Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,

    M. Oshikawa, “Topological approach to luttinger’s theorem and the fermi surface of a kondo lattice,” Physical review letters84 15(2000) 3370–3. – 25 –

    work page 2000

  20. [20]

    Lieb-Schultz-Mattis in Higher Dimensions

    M. B. Hastings, “Lieb-Schultz-Mattis in higher dimensions,” Phys. Rev. B69(2004) 104431, cond-mat/0305505

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  21. [21]

    Topological Defect Lines and Renormalization Group Flows in Two Dimensions

    C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, “Topological Defect Lines and Renormalization Group Flows in Two Dimensions,” JHEP01(2019) 026,1802.04445

    work page Pith review arXiv 2019

  22. [22]

    Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems,

    X.-G. Wen, “Emergent anomalous higher symmetries from topological order and from dynamical electromagnetic field in condensed matter systems,” Phys. Rev. B99(2019), no. 20, 205139,1812.02517

    work page arXiv 2019

  23. [23]

    Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,

    R. Thorngren and Y. Wang, “Fusion category symmetry. Part I. Anomaly in-flow and gapped phases,” JHEP04(2024) 132,1912.02817

    work page arXiv 2024

  24. [24]

    Selfenergy and Flavor Interpretation of Staggered Fermions,

    M. F. L. Golterman and J. Smit, “Selfenergy and Flavor Interpretation of Staggered Fermions,” Nucl. Phys. B245(1984) 61–88

    work page 1984

  25. [25]

    STAGGERED MESONS,

    M. F. L. Golterman, “STAGGERED MESONS,” Nucl. Phys. B273(1986) 663–676

    work page 1986

  26. [26]

    A Tool Kit for Staggered Fermions,

    G. W. Kilcup and S. R. Sharpe, “A Tool Kit for Staggered Fermions,” Nucl. Phys. B283 (1987) 493–550

    work page 1987

  27. [27]

    Lattice Fermions in Euclidean Space-time,

    L. H. Karsten, “Lattice Fermions in Euclidean Space-time,” Phys. Lett. B104(1981) 315–319

    work page 1981

  28. [28]

    ON LATTICE FERMIONS,

    F. Wilczek, “ON LATTICE FERMIONS,” Phys. Rev. Lett.59(1987) 2397

    work page 1987

  29. [29]

    Four-dimensional graphene and chiral fermions

    M. Creutz, “Four-dimensional graphene and chiral fermions,” JHEP04(2008) 017, 0712.1201

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  30. [30]

    Creutz Fermions on an Orthogonal Lattice

    A. Borici, “Creutz fermions on an orthogonal lattice,” Phys. Rev. D78(2008) 074504, 0712.4401

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  31. [31]

    Minimal doubling and point splitting

    M. Creutz, “Minimal doubling and point splitting,” PoSLATTICE2010(2010) 078, 1009.3154

    work page Pith review arXiv 2010

  32. [32]

    Classification of Minimally Doubled Fermions

    M. Creutz and T. Misumi, “Classification of Minimally Doubled Fermions,” Phys. Rev. D82 (2010) 074502,1007.3328

    work page Pith review arXiv 2010

  33. [33]

    Index Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions

    M. Creutz, T. Kimura, and T. Misumi, “Index Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions,” JHEP12(2010) 041,1011.0761

    work page Pith review arXiv 2010

  34. [34]

    Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,

    X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, “Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates,” Physical Review B 83(2010) 205101

    work page 2010

  35. [35]

    Minimal-doubling and single-Weyl Hamiltonians

    T. Misumi, “Minimal-doubling and single-Weyl Hamiltonians,”2512.22609

    work page internal anchor Pith review Pith/arXiv arXiv

  36. [36]

    A Method for Simulating Chiral Fermions on the Lattice

    D. B. Kaplan, “A Method for simulating chiral fermions on the lattice,” Phys. Lett. B288 (1992) 342–347,hep-lat/9206013

    work page Pith review arXiv 1992

  37. [37]

    Chiral Fermions from Lattice Boundaries

    Y. Shamir, “Chiral fermions from lattice boundaries,” Nucl. Phys. B406(1993) 90–106, hep-lat/9303005

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  38. [38]

    Spectral properties and chiral symmetry violations of (staggered) domain wall fermions in the Schwinger model,

    C. Hoelbling and C. Zielinski, “Spectral properties and chiral symmetry violations of (staggered) domain wall fermions in the Schwinger model,” Phys. Rev. D94(2016), no. 1, 014501,1602.08432

    work page arXiv 2016

  39. [39]

    Staggered domain wall fermions,

    C. Hoelbling and C. Zielinski, “Staggered domain wall fermions,” PoSLATTICE2016 (2016) 254,1609.05114. – 26 –

    work page arXiv 2016

  40. [40]

    Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions,

    A. N. Redlich, “Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions,” Phys. Rev. D29(1984) 2366–2374

    work page 1984

  41. [41]

    Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions,

    A. N. Redlich, “Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions,” Phys. Rev. Lett.52(1984) 18

    work page 1984

  42. [42]

    Gapped Boundary Phases of Topological Insulators via Weak Coupling,

    N. Seiberg and E. Witten, “Gapped Boundary Phases of Topological Insulators via Weak Coupling,” PTEP2016(2016), no. 12, 12C101,1602.04251

    work page arXiv 2016

  43. [43]

    Golterman,Staggered fermions,2406.02906

    M. Golterman, “Staggered fermions,”2406.02906

    work page arXiv

  44. [44]

    C-R-T Fractionalization, Fermions, and Mod 8 Periodicity,

    Z. Wan, J. Wang, S.-T. Yau, and Y.-Z. You, “C-R-T Fractionalization, Fermions, and Mod 8 Periodicity,”2312.17126

    work page arXiv

  45. [45]

    C-R-T Fractionalization in the First Quantized Hamiltonian Theory

    Y.-Y. Li, Z. Wan, J. Wang, S.-T. Yau, and Y.-Z. You, “C-R-T Fractionalization in the First Quantized Hamiltonian Theory,”2412.11958

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [46]

    Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,

    M. Cheng and N. Seiberg, “Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,” SciPost Phys.15(2023), no. 2, 051,2211.12543

    work page arXiv 2023

  47. [47]

    Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,

    N. Seiberg and S.-H. Shao, “Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries,” SciPost Phys.16(2024), no. 3, 064,2307.02534

    work page arXiv 2024

  48. [48]

    Disclinations, Dislocations, and Emanant Flux at Dirac Criticality,

    M. Barkeshli, C. Fechisin, Z. Komargodski, and S. Zhong, “Disclinations, Dislocations, and Emanant Flux at Dirac Criticality,” Phys. Rev. X16(2026), no. 1, 011017,2501.13866

    work page arXiv 2026

  49. [49]

    Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice,

    M. Oshikawa, “Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice,” Phys. Rev. Lett.84(2000), no. 7, 1535

    work page 2000

  50. [50]

    Anomalies and Fermion Zero Modes on Strings and Domain Walls,

    C. G. Callan, Jr. and J. A. Harvey, “Anomalies and Fermion Zero Modes on Strings and Domain Walls,” Nucl. Phys. B250(1985) 427–436

    work page 1985

  51. [51]

    A one-way quantum computer,

    R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86 (2001) 5188–5191

    work page 2001

  52. [52]

    Classification of gapped symmetric phases in one-dimensional spin systems,

    X. Chen, Z.-C. Gu, and X.-G. Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Phys. Rev. B83(2011) 035107

    work page 2011

  53. [53]

    Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator,

    H. Fukaya, T. Onogi, and S. Yamaguchi, “Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator,” Phys. Rev. D96(2017), no. 12, 125004,1710.03379

    work page arXiv 2017

  54. [54]

    Physicist-Friendly Reformulation of the Atiyah-Patodi-Singer Index on a Lattice with Boundary,

    H. Fukaya, M. Furuta, S. Matsuo, T. Onogi, S. Yamaguchi, and M. Yamashita, “Physicist-Friendly Reformulation of the Atiyah-Patodi-Singer Index on a Lattice with Boundary,” Commun. Math. Phys.380(2020), no. 3, 1295–1311,1910.01987

    work page arXiv 2020

  55. [55]

    A Remnant of Chiral Symmetry on the Lattice,

    P. H. Ginsparg and K. G. Wilson, “A Remnant of Chiral Symmetry on the Lattice,” Phys. Rev. D25(1982) 2649

    work page 1982

  56. [56]

    Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation

    M. Luscher, “Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation,” Phys. Lett. B428(1998) 342–345,hep-lat/9802011

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  57. [57]

    More about exactly massless quarks on the lattice

    H. Neuberger, “More about exactly massless quarks on the lattice,” Phys. Lett. B427 (1998) 353–355,hep-lat/9801031

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  58. [58]

    Gapped Chiral Fermions,

    S. S. Razamat and D. Tong, “Gapped Chiral Fermions,” Phys. Rev. X11(2021), no. 1, 011063,2009.05037

    work page arXiv 2021

  59. [59]

    Tong,Comments on symmetric mass generation in 2d and 4d,JHEP07(2022) 001, [arXiv:2104.03997]

    D. Tong, “Comments on symmetric mass generation in 2d and 4d,” JHEP07(2022) 001, 2104.03997

    work page arXiv 2022

  60. [60]

    Staggered Fermions with Chiral Anomaly Cancellation,

    L.-X. Xu, “Staggered Fermions with Chiral Anomaly Cancellation,”2501.10837. – 27 –

    work page arXiv

  61. [61]

    Theoretical foundation for the index theorem on the lattice with staggered fermions,

    D. H. Adams, “Theoretical foundation for the index theorem on the lattice with staggered fermions,” Phys.Rev.Lett.104(2010) 141602,0912.2850

    work page arXiv 2010

  62. [62]

    Pairs of chiral quarks on the lattice from staggered fermions,

    D. H. Adams, “Pairs of chiral quarks on the lattice from staggered fermions,” Phys.Lett.B 699(2011) 394–397,1008.2833

    work page arXiv 2011

  63. [63]

    Single flavor staggered fermions,

    C. Hoelbling, “Single flavor staggered fermions,” Phys.Lett.B696(2011) 422–425, 1009.5362

    work page arXiv 2011

  64. [64]

    Aoki Phases in the Lattice Gross-Neveu Model with Flavored Mass terms,

    M. Creutz, T. Kimura, and T. Misumi, “Aoki Phases in the Lattice Gross-Neveu Model with Flavored Mass terms,” Phys. Rev. D83(2011) 094506,1101.4239

    work page arXiv 2011

  65. [65]

    Numerical properties of staggered quarks with a taste-dependent mass term,

    P. de Forcrand, A. Kurkela, and M. Panero, “Numerical properties of staggered quarks with a taste-dependent mass term,” JHEP04(2012) 142,1202.1867

    work page arXiv 2012

  66. [66]

    Strong-coupling analysis of parity phase structure in staggered-wilson fermions,

    T. Misumi, T. Z. Nakano, T. Kimura, and A. Ohnishi, “Strong-coupling analysis of parity phase structure in staggered-wilson fermions,” Phys.Rev.D86(2012) 034501,1205.6545

    work page arXiv 2012

  67. [67]

    New fermion discretizations and their applications

    T. Misumi, “New fermion discretizations and their applications,” PoSLATTICE2012 (2012) 005,1211.6999

    work page Pith review arXiv 2012

  68. [68]

    Varieties and properties of central-branch Wilson fermions,

    T. Misumi and J. Yumoto, “Varieties and properties of central-branch Wilson fermions,” Phys. Rev. D102(2020), no. 3, 034516,2005.08857

    work page arXiv 2020

  69. [69]

    Symmetry properties of staggered fermions with taste splitting mass term,

    N. Chreim and C. Hoelbling, “Symmetry properties of staggered fermions with taste splitting mass term,” PoSLATTICE2024(2025) 370,2411.07780

    work page arXiv 2025

  70. [70]

    Revisiting symmetries of lattice fermions via spin-flavor representation,

    T. Kimura, S. Komatsu, T. Misumi, T. Noumi, S. Torii, and S. Aoki, “Revisiting symmetries of lattice fermions via spin-flavor representation,” JHEP01(2012) 048,1111.0402. – 28 –

    work page arXiv 2012