Journal of High Energy Physics

, 2010:41 | Cite as

Index theorem and overlap formalism with naive and minimally doubled fermions

  • Michael Creutz
  • Taro KimuraEmail author
  • Tatsuhiro Misumi


We present a theoretical foundation for the Index theorem in naive and minimally doubled lattice fermions by studying the spectral flow of a Hermitean version of Dirac operators. We utilize the point splitting method to implement flavored mass terms, which play an important role in constructing proper Hermitean operators. We show the spectral flow correctly detects the index of the would-be zero modes which is determined by gauge field topology. Using the flavored mass terms, we present new types of overlap fermions from the naive fermion kernels, with a number of flavors that depends on the choice of the mass terms. We succeed to obtain a single-flavor naive overlap fermion which maintains hypercubic symmetry.


Lattice Gauge Field Theories Lattice Quantum Field Theory Lattice QCD 


  1. [1]
    K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445 [SPIRES].ADSGoogle Scholar
  2. [2]
    J. Smit and J.C. Vink, Remnants of the Index Theorem on the Lattice, Nucl. Phys. B 286 (1987) 485 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    P.H. Ginsparg and K.G. Wilson, A Remnant of Chiral Symmetry on the Lattice, Phys. Rev. D 25 (1982) 2649 [SPIRES].ADSGoogle Scholar
  4. [4]
    H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett. B 427 (1998) 353 [hep-lat/9801031] [SPIRES].ADSGoogle Scholar
  5. [5]
    R.G. Edwards, U.M. Heller and R. Narayanan, The hermitian Wilson-Dirac operator in smooth SU(2) instanton backgrounds, Nucl. Phys. B 522 (1998) 285 [hep-l at / 9801015] [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    D. H. Adams, Axial anomaly and topological charge in lattice gauge theory with overlap-Dirac operator, Annals Phys. 296 (2002) 131 [hep-lat/9812003] [SPIRES].CrossRefzbMATHADSGoogle Scholar
  7. [7]
    D.H. Adams, On the continuum limit of fermionic topological charge in lattice gauge theory, J. Math. Phys. 42 (2001) 5522 [hep-lat/0009026] [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  8. [8]
    D.B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B 288 (1992) 342 [hep-lat/9206013] [SPIRES].ADSGoogle Scholar
  9. [9]
    V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B 439 (1995) 54 [hep-lat/9405004] [SPIRES].CrossRefADSGoogle Scholar
  10. [10]
    J.B. Kogut and L. Susskind, Hamiltonian Formulation of Wilson’s Lattice Gauge Theories, Phys. Rev. D 11 (1975) 395 [SPIRES].ADSGoogle Scholar
  11. [11]
    L. Susskind, Lattice Fermions, Phys. Rev. D 16 (1977) 3031 [SPIRES].ADSGoogle Scholar
  12. [12]
    H.S. Sharatchandra, H.J. Thun and P. Weisz, Susskind Fermions on a Euclidean Lattice, Nucl. Phys. B 192 (1981) 205 [SPIRES].CrossRefADSGoogle Scholar
  13. [13]
    J. Smit and J.C. Vink, Renormalized Ward-Takahashi Relations and Topological Susceptibility With Staggered Fermions, Nucl. Phys. B 298 (1988) 557 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    D.H. Adams, Theoretical foundation for the Index Theorem on the lattice with staggered fermions, Phys. Rev. Lett. 104 (2010) 141602 [arXiv:0912.2850] [SPIRES].CrossRefADSGoogle Scholar
  15. [15]
    D.H. Adams, Pairs of massless quarks on the lattice from staggered fermions, arXiv:1008.2833 [SPIRES].
  16. [16]
    C. Hölbling, Single flavor staggered overlap, arXiv:1009.5362 [SPIRES].
  17. [17]
    L.H. Karsten, Lattice Fermions in Euclidean Space-Time, Phys. Lett. B 104 (1981) 315 [SPIRES].CrossRefADSGoogle Scholar
  18. [18]
    F. Wilczek, On Lattice Fermions, Phys. Rev. Lett. 59 (1987) 2397 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    M. Creutz, Four-dimensional graphene and chiral fermions, JHEP 04 (2008) 017 [arXiv:0712.1201] [SPIRES].CrossRefADSGoogle Scholar
  20. [20]
    A. Borici, Creutz fermions on an orthogonal lattice, Phys. Rev. D 78 (2008) 074504 [arXiv:0712.4401] [SPIRES].ADSGoogle Scholar
  21. [21]
    P.F. Bedaque, M.I. Buchoff, B.C. Tiburzi and A. Walker-Loud, Broken Symmetries from Minimally Doubled Fermions, Phys. Lett. B 662 (2008) 449 [arXiv:0801.3361] [SPIRES].ADSMathSciNetGoogle Scholar
  22. [22]
    P.F. Bedaque, M.I. Buchoff, B.C. Tiburzi and A. Walker-Loud, Search for Fermion Actions on Hyperdiamond Lattices, Phys. Rev. D 78 (2008) 017502 [arXiv:0804.1145] [SPIRES].ADSGoogle Scholar
  23. [23]
    S. Capitani, J. Weber and H. Wittig, Minimally doubled fermions at one loop, Phys. Lett. B 681 (2009) 105 [arXiv:0907.2825] [SPIRES].ADSMathSciNetGoogle Scholar
  24. [24]
    T. Kimura and T. Misumi, Characters of Lattice Fermions Based on the Hyperdiamond Lattice, Prog. Theor. Phys. 124 (2010) 415 [arXiv:0907.1371] [SPIRES].CrossRefzbMATHADSGoogle Scholar
  25. [25]
    T. Kimura and T. Misumi, Lattice Fermions Based on Higher-Dimensional Hyperdiamond Lattices, Prog. Theor. Phys. 123 (2010) 63 [arXiv:0907.3774] [SPIRES].CrossRefzbMATHADSGoogle Scholar
  26. [26]
    S. Capitani, M. Creutz, J. Weber and H. Wittig, Renormalization of minimally doubled fermions, JHEP 09 (2010) 027 [arXiv:1006.2009] [SPIRES].CrossRefADSGoogle Scholar
  27. [27]
    M. Creutz and T. Misumi, Classification of Minimally Doubled Fermions, Phys. Rev. D 82 (2010) 074502 [arXiv:1007.3328] [SPIRES].ADSGoogle Scholar
  28. [28]
    T. Misumi, M. Creutz and T. Kimura, Classification and Generalization of Minimal-doubling actions, arXiv:1010.3713 [SPIRES].
  29. [29]
    M. Creutz, Minimal doubling and point splitting, arXiv:1009.3154 [SPIRES].
  30. [30]
    B.C. Tiburzi, Chiral Lattice Fermions, Minimal Doubling and the Axial Anomaly, Phys. Rev. D 82 (2010) 034511 [arXiv:1006.0172] [SPIRES].ADSGoogle Scholar
  31. [31]
    C. van den Doel and J. Smit, Dynamical Symmetry Breaking in Two Flavor SU(N) and SO(N) Lattice Gauge Theories, Nucl. Phys. B 228 (1983) 122 [SPIRES].CrossRefADSGoogle Scholar
  32. [32]
    M.F.L. Golterman and J. Smit, Selfenergy and Flavor Interpretation of Staggered Fermions, Nucl. Phys. B 245 (1984) 61 [SPIRES].CrossRefADSGoogle Scholar
  33. [33]
    M.F.L. Golterman, Staggered Mesons, Nucl. Phys. B 273 (1986) 663 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  34. [34]
    M. Creutz, Anomalies, gauge field topology and the lattice, arXiv:1007.5502 [SPIRES].
  35. [35]
    J. Smit and J.C. Vink, Topological Charge And Fermions in the Two-Dimensional Lattice U(1) Model. 1. Staggered Fermions, Nucl. Phys. B 303 (1988) 36 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    J. Smit and J.C. Vink, Staggered fermions and topological susceptibility in lattice QCD at β = 5.7, Phys. Lett. B 194 (1987) 433 [SPIRES].ADSGoogle Scholar
  37. [37]
    J.C. Vink, Flavor symmetry breaking and zero mode shift for staggered fermions, Phys. Lett. B 210 (1988) 211 [SPIRES].ADSGoogle Scholar
  38. [38]
    J.C. Vink, Staggered fermions, topological charge and topological susceptibility in lattice QCD, Phys. Lett. B 212 (1988) 483 [SPIRES].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.Physics DepartmentBrookhaven National LaboratoryUptonU.S.A.
  2. 2.Department of Basic ScienceUniversity of TokyoTokyoJapan
  3. 3.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan

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