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 1998,¹ they were quickly taken up for a variety of numerical simulations.²³⁴ By now overlap fermions are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD.⁵⁶

Overlap fermions with mass ${\displaystyle m}$ are defined on a Euclidean spacetime lattice with spacing ${\displaystyle a}$ by the overlap Dirac operator

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

where ${\displaystyle A}$ is the ″kernel″ Dirac operator obeying ${\displaystyle \gamma _{5}A=A^{\dagger }\gamma _{5}}$, i.e. ${\displaystyle A}$ is ${\displaystyle \gamma _{5}}$-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations.⁷ A common choice for the kernel is

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

where ${\displaystyle D}$ is the massless Dirac operator and ${\displaystyle s\in \left(-1,1\right)}$ is a free parameter that can be tuned to optimise locality of ${\displaystyle D_{\text{ov}}}$.⁸

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

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

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

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

and decouple.

Overlap fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (obeying the Ginsparg–Wilson equation) and locality.⁹