Overlap fermion
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  are defined on a Euclidean spacetime lattice with spacing  by the overlap Dirac operator

where  is the ″kernel″ Dirac operator obeying , i.e. is -hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations. A common choice for the kernel is

where  is the massless Dirac operator and  is a free parameter that can be tuned to optimise locality of . Near  the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)

whereas the unphysical doublers near  are suppressed by a high mass

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 

Physics
Theoretical physics