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Lattice Fermions and Chirality

A local lattice Dirac operator confronts a tension between chirality and the desired particle spectrum. This chapter explains doubling and the Nielsen–Ninomiya obstruction, compares the principal fermion formulations, and then separates chiral Ward identities, determinant positivity, and the construction of a chiral gauge measure.

Enter this chapter

Section titled “Enter this chapter”

Begin with Naive Fermions and Species Doubling to locate the extra zeros of the naive lattice Dirac operator.

Topics

Section titled “Topics”
  1. Naive Fermions and Species Doubling — Counts species through the periodic momentum symbol.
  2. The Nielsen–Ninomiya Obstruction — States the hypotheses behind the doubling obstruction.
  3. Wilson and Clover Fermions — Lifts extra species and controls the resulting symmetry breaking.
  4. Staggered Fermions and Taste — Identifies residual tastes and their finite-spacing effects.
  5. Ginsparg–Wilson Symmetry and the Lattice Index — Replaces naive chiral symmetry by a modified exact relation.
  6. Overlap and Domain-Wall Fermions — Implements that relation and tests locality or residual breaking.
  7. Fermion Determinants, Pfaffians, and Measure Positivity — Determines when fermion integration supplies a positive weight.
  8. Anomalies, Ward Identities, and Chiral Diagnostics — Checks normalized chiral identities and anomaly reproduction.
  9. Chiral Gauge Theories on the Lattice — Separates anomaly cancellation from existence of a consistent measure.

How the ideas fit together

Section titled “How the ideas fit together”

Different discretizations evade the doubling obstruction by changing different assumptions. The Wilson term lifts unwanted zeros while breaking naive chiral symmetry, which introduces mass tuning and improvement. The Ginsparg–Wilson relation instead modifies the chiral transformation at finite spacing and supports an index. These structural advantages do not prove positivity of every determinant or solve the separate global measure problem of a chiral gauge theory. A formulation must therefore be tested against the property actually needed. Ginsparg and Wilson 1982, pp. 2649–2657.

Where to go next

Section titled “Where to go next”

Continue to Sampling Algorithms for Lattice Fields to define the target measure and the transition kernel that should preserve it.

References

Section titled “References”
  • Ginsparg, P. H., and Wilson, K. G. (1982). “A remnant of chiral symmetry on the lattice.” Physical Review D 25, 2649–2657. doi:10.1103/PhysRevD.25.2649.