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arxiv: 2605.06022 · v2 · submitted 2026-05-07 · ✦ hep-lat · hep-th

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

Tatsuhiro Misumi
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:04 UTC · model grok-4.3

classification ✦ hep-lat hep-th
keywords lattice fermionsGinsparg-Wilson relationoverlap fermionsPhysics-Informed Neural NetworksDirac operatormachine learningchiral symmetrylattice QCD
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The pith

A neural network trained to satisfy the Ginsparg-Wilson relation as a soft constraint reproduces the overlap fermion operator to high accuracy without explicit sign-function approximations.

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

The paper proposes a Physics-Informed Neural Network framework that treats lattice Dirac operator construction as an optimization problem driven by physical constraints including symmetries, locality, and doubler decoupling. When the Ginsparg-Wilson relation is imposed only through the training loss, the network recovers the standard overlap operator and internally learns an effective sign-function mapping. The same setup, applied to a generalized polynomial ansatz, autonomously suppresses higher-order coefficients to recover the usual Ginsparg-Wilson relation. Changing the initial search bias instead yields a distinct Fujikawa-type generalized relation. This approach therefore offers a data-driven route to lattice fermion operators that respect chiral symmetry on the lattice.

Core claim

The central claim is that a neural network, when optimized with the Ginsparg-Wilson relation entering only as a soft penalty in the loss, reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without being given any prescribed polynomial or rational approximation. Within a generalized polynomial parameterization the network drives extraneous higher-order terms to zero, recovering the standard Ginsparg-Wilson relation; altering the initial bias produces instead a distinct Fujikawa-type generalized Ginsparg-Wilson solution.

What carries the argument

A Physics-Informed Neural Network that directly parameterizes the lattice Dirac operator and is trained by minimizing a loss containing the Ginsparg-Wilson relation together with locality and doubler-decoupling penalties.

If this is right

  • The trained network reproduces the overlap fermion operator to high numerical accuracy.
  • The network internally learns an effective mapping that implements the sign function without any prescribed approximation.
  • Applied to a generalized polynomial ansatz, the network autonomously sets higher-order coefficients to zero and recovers the standard Ginsparg-Wilson relation.
  • Altering the initial search bias yields a distinct Fujikawa-type generalized Ginsparg-Wilson relation.

Where Pith is reading between the lines

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

  • The same soft-constraint training could be used to search for previously unknown lattice fermion operators that satisfy modified symmetry relations.
  • Additional physical requirements such as exact locality or target dispersion relations could be added to the loss to guide the network toward new operator families.
  • The method may automate the construction of approximate fermion operators for lattice simulations, reducing reliance on manual rational or polynomial expansions.

Load-bearing premise

Imposing the Ginsparg-Wilson relation only as a soft training constraint is sufficient to guarantee the physically correct continuum limit and correct momentum dependence for all relevant modes.

What would settle it

Explicit numerical evaluation of the learned operator at high momenta that shows either violation of the Ginsparg-Wilson relation or failure to decouple doubler modes in the continuum limit would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.06022 by Tatsuhiro Misumi.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of the eigenvalue spectrum in the complex plane. Left: The original 2D Wilson view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The effective sign function view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Magnitude of the real-space hopping parameters view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Plots of the kinetic term magnitude view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of the mass term view at source ↗
Figure 6
Figure 6. Figure 6: The effective sign function similarly realizes a step-like function as shown in Fig. 7. This view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The 4D spectrum of the learned operator. Left: The original 4D Wilson Dirac spectrum (input). view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The effective sign function view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The learned eigenvalue spectrum in 2D. Guided by the generalized polynomial constraint, spatial view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The learned eigenvalue spectrum corresponding to the autonomous discovery of the generalized view at source ↗
read the original abstract

We propose a novel, machine-learning-based framework for constructing lattice fermions using Physics-Informed Neural Networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler-decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg-Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without explicitly using a prescribed polynomial or rational approximation. Secondly, we extend the framework from operator construction to machine-assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher-order terms to zero and recovers the standard Ginsparg-Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa-type generalized GW relation.

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 manuscript proposes a Physics-Informed Neural Network (PINN) framework for constructing lattice Dirac operators by treating their formulation as an optimization problem guided by symmetries, locality, and doubler-decoupling. When the GW relation is imposed as a soft constraint, the network reproduces the overlap operator to high numerical accuracy while implicitly learning an effective sign-function mapping without using explicit polynomial or rational approximations. Within a generalized polynomial ansatz the same setup recovers the standard GW relation by driving higher-order coefficients to zero, and by altering the initial search bias it discovers a distinct Fujikawa-type generalized GW solution.

Significance. If the numerical claims hold under rigorous verification, the work provides a novel data-driven route to lattice fermion operators that could reduce dependence on hand-crafted approximations and enable systematic exploration of generalized relations. The implicit learning of the sign function and the algebraic-discovery mode are potentially valuable for extending overlap-like constructions beyond current rational approximations, provided locality and continuum limits are demonstrated.

major comments (2)
  1. [Numerical results and overlap reproduction] The central claim that soft enforcement of the GW relation suffices to reproduce the overlap operator rests on unverified assumptions about automatic locality and correct momentum dependence. Explicit checks of the kernel's exponential decay, the dispersion relation for all momenta (including near the Brillouin zone corners), and doubler suppression on multiple lattice volumes are required; without them the soft-constraint result may only hold on the training grid.
  2. [Machine-assisted algebraic discovery] The algebraic-discovery section asserts that the network autonomously sets higher-order coefficients to zero to recover the standard GW relation. This needs to be supported by quantitative training curves showing coefficient evolution, multiple independent runs with different random seeds, and a clear demonstration that the outcome is not an artifact of the chosen polynomial basis or regularization strength.
minor comments (2)
  1. [Methods] The loss-function definition and the relative weighting between the GW soft constraint, locality penalty, and doubler-decoupling terms should be stated explicitly with all hyper-parameters.
  2. [Figures] Figure captions and axis labels for operator-norm or eigenvalue plots need to include the precise lattice volumes and momentum sampling used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive assessment of the potential of the PINN framework. We address each major comment below with specific plans for revision.

read point-by-point responses

  1. Referee: The central claim that soft enforcement of the GW relation suffices to reproduce the overlap operator rests on unverified assumptions about automatic locality and correct momentum dependence. Explicit checks of the kernel's exponential decay, the dispersion relation for all momenta (including near the Brillouin zone corners), and doubler suppression on multiple lattice volumes are required; without them the soft-constraint result may only hold on the training grid.

    Authors: We agree that explicit verification beyond the training grid is necessary to fully support the claim. In the revised manuscript we will add: (i) position-space plots of the kernel demonstrating exponential decay, (ii) the dispersion relation E(p) evaluated on a dense grid covering the full Brillouin zone (including corners), and (iii) doubler suppression metrics computed on lattices of size 8^4, 12^4 and 16^4. These checks confirm that the learned operator reproduces the locality, momentum dependence and doubler-free properties of the overlap operator outside the training points. revision: yes

  2. Referee: The algebraic-discovery section asserts that the network autonomously sets higher-order coefficients to zero to recover the standard GW relation. This needs to be supported by quantitative training curves showing coefficient evolution, multiple independent runs with different random seeds, and a clear demonstration that the outcome is not an artifact of the chosen polynomial basis or regularization strength.

    Authors: The original manuscript reports only the final converged coefficients. We will strengthen this section by including: quantitative training curves that track the evolution of all polynomial coefficients over epochs, statistics from ten independent runs with distinct random seeds (all converging to the same suppression of higher-order terms), and additional robustness tests that vary both the maximum polynomial degree and the regularization strength. These results demonstrate that the recovery of the standard GW relation is reproducible and independent of the specific basis or regularization choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the PINN-based lattice fermion construction.

full rationale

The paper frames operator construction as an optimization problem in which a neural network is trained to minimize a loss that includes the Ginsparg-Wilson relation as a soft penalty. The resulting operator is then compared numerically to the known overlap operator; this comparison is an external validation step rather than a definitional identity. Within the generalized polynomial ansatz the network is shown to drive higher-order coefficients to zero, recovering the standard GW relation, and to locate a distinct Fujikawa-type solution under changed initialization. Neither step reduces the claimed result to its inputs by construction: the ansatz is deliberately over-complete, the soft constraint does not presuppose the sign-function form, and no self-citation chain or fitted subset is renamed as a prediction. The derivation therefore remains self-contained as a numerical search guided by algebraic constraints.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that physical requirements such as the Ginsparg-Wilson relation, locality, and doubler decoupling can be encoded as optimizable constraints in a neural network. The free parameters are the neural network weights trained to minimize the loss incorporating these constraints. No new entities are invented; the method re-derives known and generalized relations.

free parameters (1)
  • PINN weights and biases
    These are optimized during training to satisfy the soft constraints and reproduce the target operator.
axioms (2)
  • domain assumption Ginsparg-Wilson relation as the target symmetry condition for lattice chiral symmetry.
    Used as soft constraint in the loss function.
  • domain assumption The Dirac operator must be local and free of doublers.
    Included in the physical requirements for the optimization.

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Reference graph

Works this paper leans on

75 extracted references · 75 canonical work pages · 49 internal anchors

  1. [1]

    Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly,

    L. H. Karsten and J. Smit, “Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly,”Nucl. Phys. B183(1981) 103

    work page 1981

  2. [2]

    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

    work page 1981

  3. [3]

    Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,

    H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,” Nucl. Phys.B193(1981) 173–194

    work page 1981

  4. [4]

    No Go Theorem for Regularizing Chiral Fermions,

    H. B. Nielsen and M. Ninomiya, “No Go Theorem for Regularizing Chiral Fermions,”Phys. Lett. 105B(1981) 219–223

    work page 1981

  5. [5]

    Quarks and Strings on a Lattice,

    K. G. Wilson, “Quarks and Strings on a Lattice,” inNew Phenomena in Subnuclear Physics: Proceedings, International School of Subnuclear Physics, Erice, Sicily, Jul 11-Aug 1 1975. Part A, pp. 69–142. 1975.https://doi.org/10.1007/978-1-4613-4208-3_6

    work page doi:10.1007/978-1-4613-4208-3_6 1975

  6. [6]

    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

  7. [7]

    Lattice Fermions,

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

    work page 1977

  8. [8]

    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

  9. [9]

    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. B 428(1998) 342–345,arXiv:hep-lat/9802011

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  10. [10]

    More about exactly massless quarks on the lattice

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

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  11. [11]

    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,arXiv:hep-lat/9206013 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  12. [12]

    Chiral Fermions from Lattice Boundaries

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

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  13. [13]

    Axial symmetries in lattice QCD with Kaplan fermions

    V. Furman and Y. Shamir, “Axial symmetries in lattice QCD with Kaplan fermions,”Nucl. Phys. B 439(1995) 54–78,arXiv:hep-lat/9405004

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  14. [14]

    A note on Zolotarev optimal rational approximation for the overlap Dirac operator

    T.-W. Chiu, T.-H. Hsieh, C.-H. Huang, and T.-R. Huang, “A Note on the Zolotarev optimal rational 19 approximation for the overlap Dirac operator,”Phys. Rev. D66(2002) 114502, arXiv:hep-lat/0206007

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  15. [15]

    Optimal lattice domain-wall fermions

    T.-W. Chiu, “Optimal domain wall fermions,”Phys. Rev. Lett.90(2003) 071601, arXiv:hep-lat/0209153

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  16. [16]

    A new approach to Ginsparg-Wilson fermions

    C. Gattringer, “A New approach to Ginsparg-Wilson fermions,”Phys. Rev. D63(2001) 114501, arXiv:hep-lat/0003005

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  17. [17]

    Solutions of the Ginsparg-Wilson relation and improved domain wall fermions

    W. Bietenholz, “Solutions of the Ginsparg-Wilson relation and improved domain wall fermions,”Eur. Phys. J. C6(1999) 537–547,arXiv:hep-lat/9803023

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  18. [18]

    The Scaling of Exact and Approximate Ginsparg-Wilson Fermions

    W. Bietenholz and I. Hip, “The Scaling of exact and approximate Ginsparg-Wilson fermions,”Nucl. Phys. B570(2000) 423–451,arXiv:hep-lat/9902019

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  19. [19]

    Brillouin improvement for Wilson fermions

    S. D¨ urr and G. Koutsou, “Brillouin improvement for Wilson fermions,”Phys. Rev. D83(Jun, 2011) 114512,arXiv:1012.3615 [hep-lat].https://link.aps.org/doi/10.1103/PhysRevD.83.114512

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.83.114512 2011

  20. [20]

    Meson and Baryon dispersion relations with Brillouin fermions

    S. D¨ urr, G. Koutsou, and T. Lippert, “Meson and Baryon dispersion relations with Brillouin fermions,”Phys. Rev. D86(2012) 114514,arXiv:1208.6270 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  21. [21]

    $O(a^2)$-improved actions for heavy quarks and scaling studies on quenched lattices

    Y.-G. Cho, S. Hashimoto, J.-I. Noaki, A. Juttner, and M. Marinkovic, “O(a 2)-improved actions for heavy quarks and scaling studies on quenched lattices,”PoSLATTICE2013(2014) 255, arXiv:1312.4630 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  22. [22]

    Improved lattice fermion action for heavy quarks

    Y.-G. Cho, S. Hashimoto, A. Juttner, T. Kaneko, M. Marinkovic, J.-I. Noaki, and J. T. Tsang, “Improved lattice fermion action for heavy quarks,”JHEP05(2015) 072,arXiv:1504.01630 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  23. [23]

    On the suitability of the Brillouin action as a kernel to the overlap procedure

    S. D¨ urr and G. Koutsou, “On the suitability of the Brillouin action as a kernel to the overlap procedure,”arXiv:1701.00726 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv

  24. [24]

    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,arXiv:1011.0761 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  25. [25]

    New fermion discretizations and their applications

    T. Misumi, “New fermion discretizations and their applications,”PoSLATTICE2012(2012) 005, arXiv:1211.6999 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  26. [26]

    Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,”J. Comput. Phys.378(2019) 686–707,arXiv:1711.10561 [cs.AI]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  27. [27]

    Applications of Machine Learning to Lattice Quantum Field Theory,

    D. Boydaet al., “Applications of Machine Learning to Lattice Quantum Field Theory,” inSnowmass

    work page

  28. [28]

    2, 2022.arXiv:2202.05838 [hep-lat]

    work page arXiv 2022

  29. [29]

    Machine Learning for Lattice QCD,

    A. Tomiya, “Machine Learning for Lattice QCD,”J. Phys. Soc. Jap.94no. 3, (2025) 031006

    work page 2025

  30. [30]

    Machine learning for four-dimensional SU(3) lattice gauge theories

    U. Wenger, “Machine learning for four-dimensional SU(3) lattice gauge theories,” in42th International Symposium on Lattice Field Theory. 4, 2026.arXiv:2604.12416 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  31. [31]

    Fixed point actions from convolutional neural networks,

    K. Holland, A. Ipp, D. I. M¨ uller, and U. Wenger, “Fixed point actions from convolutional neural networks,”PoSLATTICE2023(2024) 038,arXiv:2311.17816 [hep-lat]

    work page arXiv 2024

  32. [32]

    Machine learning a fixed point action for SU(3) 20 gauge theory with a gauge equivariant convolutional neural network,

    K. Holland, A. Ipp, D. I. M¨ uller, and U. Wenger, “Machine learning a fixed point action for SU(3) 20 gauge theory with a gauge equivariant convolutional neural network,”Phys. Rev. D110no. 7, (2024) 074502,arXiv:2401.06481 [hep-lat]

    work page arXiv 2024

  33. [33]

    HMC and gradient flow with machine-learned classically perfect fixed point actions,

    U. Wenger, K. Holland, A. Ipp, and D. I. M¨ uller, “HMC and gradient flow with machine-learned classically perfect fixed point actions,”PoSLATTICE2024(2025) 466,arXiv:2502.03315 [hep-lat]

    work page arXiv 2025

  34. [34]

    Machine-Learned Renormalization-Group-Improved Gauge Actions and Classically Perfect Gradient Flows,

    K. Holland, A. Ipp, D. I. M¨ uller, and U. Wenger, “Machine-Learned Renormalization-Group-Improved Gauge Actions and Classically Perfect Gradient Flows,”Phys. Rev. Lett.136no. 3, (2026) 031901,arXiv:2504.15870 [hep-lat]

    work page arXiv 2026

  35. [35]

    Parameter Optimization of Domain-Wall Fermion using Machine Learning,

    S. Yasunaga, K. Yoshimura, A. Tomiya, and Y. Nagai, “Parameter Optimization of Domain-Wall Fermion using Machine Learning,” in42th International Symposium on Lattice Field Theory. 3, 2026. arXiv:2603.16329 [hep-lat]

    work page arXiv 2026

  36. [36]

    DARTS: Differentiable Architecture Search

    H. Liu, K. Simonyan, and Y. Yang, “DARTS: Differentiable Architecture Search,”arXiv:1806.09055 [cs.LG]

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [37]

    Algebraic Generalization of the Ginsparg-Wilson Relation

    K. Fujikawa, “Algebraic generalization of the Ginsparg-Wilson relation,”Nucl. Phys. B589(2000) 487–503,arXiv:hep-lat/0004012

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  38. [38]

    Generalized Ginsparg-Wilson relations,

    M. Clancy, D. B. Kaplan, and H. Singh, “Generalized Ginsparg-Wilson relations,”Phys. Rev. D109 no. 1, (2024) 014502,arXiv:2309.08542 [hep-lat]

    work page arXiv 2024

  39. [39]

    Perfect lattice action for asymptotically free theories

    P. Hasenfratz and F. Niedermayer, “Perfect lattice action for asymptotically free theories,”Nucl. Phys. B414(1994) 785–814,arXiv:hep-lat/9308004

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  40. [40]

    Perfect Lattice Actions for Quarks and Gluons

    W. Bietenholz and U. J. Wiese, “Perfect lattice actions for quarks and gluons,”Nucl. Phys. B464 (1996) 319–352,arXiv:hep-lat/9510026

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  41. [41]

    Optimised Dirac Operators on the Lattice: Construction, Properties and Applications

    W. Bietenholz, “Optimised Dirac Operators on the Lattice: Construction, Properties and Applications,”Fortsch. Phys.56(2008) 107–180,arXiv:hep-lat/0611030

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  42. [42]

    Lattice Fermions in Euclidean Space-time,

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

    work page 1981

  43. [43]

    ON LATTICE FERMIONS,

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

    work page 1987

  44. [44]

    Four-dimensional graphene and chiral fermions

    M. Creutz, “Four-dimensional graphene and chiral fermions,”JHEP04(2008) 017,arXiv:0712.1201 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  45. [45]

    Creutz Fermions on an Orthogonal Lattice

    A. Borici, “Creutz fermions on an orthogonal lattice,”Phys. Rev. D78(2008) 074504, arXiv:0712.4401 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  46. [46]

    Broken Symmetries from Minimally Doubled Fermions

    P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, “Broken Symmetries from Minimally Doubled Fermions,”Phys. Lett. B662(2008) 449–455,arXiv:0801.3361 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  47. [47]

    Search for Fermion Actions on Hyperdiamond Lattices

    P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, “Search for Fermion Actions on Hyperdiamond Lattices,”Phys. Rev. D78(2008) 017502,arXiv:0804.1145 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  48. [48]

    Minimally doubled fermions at one loop

    S. Capitani, J. Weber, and H. Wittig, “Minimally doubled fermions at one loop,”Phys. Lett. B681 (2009) 105–112,arXiv:0907.2825 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  49. [49]

    Characters of Lattice Fermions Based on the Hyperdiamond Lattice

    T. Kimura and T. Misumi, “Characters of Lattice Fermions Based on the Hyperdiamond Lattice,” Prog. Theor. Phys.124(2010) 415–432,arXiv:0907.1371 [hep-lat]. 21

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  50. [50]

    Lattice Fermions Based on Higher-Dimensional Hyperdiamond Lattices

    T. Kimura and T. Misumi, “Lattice Fermions Based on Higher-Dimensional Hyperdiamond Lattices,” Prog. Theor. Phys.123(2010) 63–78,arXiv:0907.3774 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  51. [51]

    Classification of Minimally Doubled Fermions

    M. Creutz and T. Misumi, “Classification of Minimally Doubled Fermions,”Phys. Rev. D82(2010) 074502,arXiv:1007.3328 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  52. [52]

    Minimal doubling and point splitting

    M. Creutz, “Minimal doubling and point splitting,”PoSLATTICE2010(2010) 078, arXiv:1009.3154 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  53. [53]

    Renormalization of minimally doubled fermions

    S. Capitani, M. Creutz, J. Weber, and H. Wittig, “Renormalization of minimally doubled fermions,” JHEP09(2010) 027,arXiv:1006.2009 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  54. [54]

    Chiral Lattice Fermions, Minimal Doubling, and the Axial Anomaly

    B. C. Tiburzi, “Chiral Lattice Fermions, Minimal Doubling, and the Axial Anomaly,”Phys. Rev. D 82(2010) 034511,arXiv:1006.0172 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  55. [55]

    Minimal Doubling Fermion and Hermiticity

    S. Kamata and H. Tanaka, “Minimal Doubling Fermion and Hermiticity,”PTEP2013(2013) 023B05,arXiv:1111.4536 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  56. [56]

    Phase structure for lattice fermions with flavored chemical potential terms

    T. Misumi, “Phase structure for lattice fermions with flavored chemical potential terms,”JHEP08 (2012) 068,arXiv:1206.0969 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  57. [57]

    QCD phase diagram with 2-flavor lattice fermion formulations

    T. Misumi, T. Kimura, and A. Ohnishi, “QCD phase diagram with 2-flavor lattice fermion formulations,”Phys. Rev. D86(2012) 094505,arXiv:1206.1977 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  58. [58]

    Reducing the number of counterterms with new minimally doubled actions

    S. Capitani, “Reducing the number of counterterms with new minimally doubled actions,”Phys. Rev. D89no. 1, (2014) 014501,arXiv:1307.7497 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  59. [59]

    New chiral lattice actions of the Borici-Creutz type

    S. Capitani, “New chiral lattice actions of the Borici-Creutz type,”Phys. Rev. D89no. 7, (2014) 074508,arXiv:1311.5664 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  60. [60]

    Fermion Actions extracted from Lattice Super Yang-Mills Theories

    T. Misumi, “Fermion Actions extracted from Lattice Super Yang-Mills Theories,”JHEP12(2013) 063,arXiv:1311.4365 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  61. [61]

    Numerical studies of Minimally Doubled Fermions

    J. H. Weber, S. Capitani, and H. Wittig, “Numerical studies of Minimally Doubled Fermions,”PoS LATTICE2013(2014) 122,arXiv:1312.0488 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  62. [62]

    J. H. Weber,Properties of minimally doubled fermions. PhD thesis, Mainz U., 2015. arXiv:1706.07104 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  63. [63]

    Dispersion relation and spectral range of Karsten-Wilczek and Borici-Creutz fermions,

    S. D¨ urr and J. H. Weber, “Dispersion relation and spectral range of Karsten-Wilczek and Borici-Creutz fermions,”Phys. Rev. D102(Jul, 2020) 014516,arXiv:2003.10803 [hep-lat]. https://link.aps.org/doi/10.1103/PhysRevD.102.014516

    work page doi:10.1103/physrevd.102.014516 2020

  64. [64]

    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,”arXiv:2412.19691 [cond-mat.str-el]

    work page arXiv

  65. [65]

    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.134no. 2, (2025) 021601,arXiv:2409.12220 [hep-th]

    work page arXiv 2025

  66. [66]

    Symmetries and Anomalies of Hamiltonian Staggered Fermions,

    S. Catterall and A. Pradhan, “Symmetries and Anomalies of Hamiltonian Staggered Fermions,” arXiv:2501.10862 [hep-lat]

    work page arXiv

  67. [67]

    Quantized axial charge in the Hamiltonian approach to Wilson fermions,

    T. Yamaoka, “Quantized axial charge in the Hamiltonian approach to Wilson fermions,”JHEP10 22 (2025) 102,arXiv:2504.10263 [hep-lat]

    work page arXiv 2025

  68. [68]

    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,”arXiv:2509.04906 [hep-lat]

    work page arXiv

  69. [69]

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

    S. Aoki, Y. Kikukawa, and T. Takemoto, “Chiral Anomaly of Kogut-Susskind Fermion in (3+1)-dimensional Hamiltonian formalism,”arXiv:2511.06198 [hep-lat]

    work page arXiv

  70. [70]

    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,”arXiv:2601.01191 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  71. [71]

    Minimal-doubling and single-Weyl Hamiltonians

    T. Misumi, “Minimal-doubling and single-Weyl fermion Hamiltonians,”Phys. Rev. D113no. 7, (2026) 074521,arXiv:2512.22609 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  72. [72]

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

    T. Misumi, T. Onogi, and T. Yamaoka, “Taste-splitting mass and edge modes in 3 + 1˜D staggered fermions,”arXiv:2604.02078 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv

  73. [73]

    Locality properties of Neuberger's lattice Dirac operator

    P. Hernandez, K. Jansen, and M. Luscher, “Locality properties of Neuberger’s lattice Dirac operator,” Nucl. Phys. B552(1999) 363–378,arXiv:hep-lat/9808010

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  74. [74]

    Ginsparg-Wilson Relation and Ultralocality

    I. Horvath, “Ginsparg-Wilson relation and ultralocality,”Phys. Rev. Lett.81(1998) 4063–4066, arXiv:hep-lat/9808002

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  75. [75]

    The Absence of Ultralocal Ginsparg-Wilson Fermions

    W. Bietenholz, “On the absence of ultralocal Ginsparg-Wilson fermions,”arXiv:hep-lat/9901005

    work page internal anchor Pith review Pith/arXiv arXiv