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}} = \frac1a \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} = \frac1a+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.