Lattice fermion discretisation
In lattice field theory, overlap fermions are a fermion discretization that allows to avoid the fermion doubling problem. They are a realisation of Ginsparg–Wilson fermions.
Initially introduced by Neuberger in 1998, they were quickly taken up for a variety of numerical simulations. By now overlap fermions are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD.
Overlap fermions with mass formula_1 are defined on a Euclidean spacetime lattice with spacing formula_2 by the overlap Dirac operator
formula_3
where formula_4 is the ″kernel″ Dirac operator obeying formula_5, i.e. formula_4 is formula_7-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations. A common choice for the kernel is
formula_8
where formula_9 is the massless Dirac operator and formula_10 is a free parameter that can be tuned to optimise locality of formula_11.
Near formula_12 the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)
formula_13
whereas the unphysical doublers near formula_14 are suppressed by a high mass
formula_15
and decouple.
Overlap fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (obeying the Ginsparg–Wilson equation) and locality.
References.
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