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 m {\displaystyle m} are defined on a Euclidean spacetime lattice with spacing a {\displaystyle a} by the overlap Dirac operator
D ov = 1 a ( ( 1 + a m ) 1 + ( 1 − a m ) γ 5 s i g n [ γ 5 A ] ) {\displaystyle D_{\text{ov}}={\frac {1}{a}}\left(\left(1+am\right)\mathbf {1} +\left(1-am\right)\gamma _{5}\mathrm {sign} [\gamma _{5}A]\right)\,}
where A {\displaystyle A} is the ″kernel″ Dirac operator obeying γ 5 A = A † γ 5 {\displaystyle \gamma _{5}A=A^{\dagger }\gamma _{5}} , i.e. A {\displaystyle A} is γ 5 {\displaystyle \gamma _{5}} -hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations. A common choice for the kernel is
A = a D − 1 ( 1 + s ) {\displaystyle A=aD-\mathbf {1} (1+s)\,}
where D {\displaystyle D} is the massless Dirac operator and s ∈ ( − 1 , 1 ) {\displaystyle s\in \left(-1,1\right)} is a free parameter that can be tuned to optimise locality of D ov {\displaystyle D_{\text{ov}}} . Near p a = 0 {\displaystyle pa=0} the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)
D ov = m + i p / 1 1 + s + O ( a ) {\displaystyle D_{\text{ov}}=m+i\,{p\!\!\!/}{\frac {1}{1+s}}+{\mathcal {O}}(a)\,}
whereas the unphysical doublers near p a = π {\displaystyle pa=\pi } are suppressed by a high mass
D ov = 1 a + m + i p / 1 1 − s + O ( a ) {\displaystyle D_{\text{ov}}={\frac {1}{a}}+m+i\,{p\!\!\!/}{\frac {1}{1-s}}+{\mathcal {O}}(a)}
and decouple. Overlap fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (obeying the Ginsparg–Wilson equation) and locality.
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