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 $m$ are defined on a Euclidean spacetime lattice with spacing $a$ by the overlap Dirac operator

$D_{\text{ov}}=\frac{1}{a}((1+am)\mathrm{𝟏}+(1-am){\gamma }_5\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}[{\gamma }_{5}A])$

where $A$ is the ″kernel″ Dirac operator obeying ${\gamma }_{5}A=A^{†}{\gamma }_5$, i.e. $A$ is ${\gamma }_5$-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations. A common choice for the kernel is

$A=aD-\mathrm{𝟏}(1+s)$

where $D$ is the massless Dirac operator and $s\in (-1,1)$ is a free parameter that can be tuned to optimise locality of $D_{\text{ov}}$.

Near $pa=0$ the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)

$D_{\text{ov}}=m+ip/\frac{1}{1+s}+𝒪(a)$

whereas the unphysical doublers near $pa=\pi$ are suppressed by a high mass

$D_{\text{ov}}=\frac{1}{a}+m+ip/\frac{1}{1-s}+𝒪(a)$

and decouple.

Overlap fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (obeying the Ginsparg–Wilson equation) and locality.