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

$D_{\text{ov}}={\frac {1}{a}}\left(\left(1+am\right)\mathbf {1} +\left(1-am\right)\gamma _{5}\mathrm {sign} [\gamma _{5}A]\right)\,$

where $A$ is the ″kernel″ Dirac operator obeying $\gamma _{5}A=A^{\dagger }\gamma _{5}$, 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

$A=aD-\mathbf {1} (1+s)\,$

where $D$ is the massless Dirac operator and $s\in \left(-1,1\right)$ 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)

$D_{\text{ov}}=m+i\,{p\!\!\!/}{\frac {1}{1+s}}+{\mathcal {O}}(a)\,$

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

$D_{\text{ov}}={\frac {1}{a}}+m+i\,{p\!\!\!/}{\frac {1}{1-s}}+{\mathcal {O}}(a)$

and decouple.

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