Overlap fermion explained

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.^[2] ^[3] ^[4] By now overlap fermions are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD.^[5] ^[6]

Overlap fermions with mass

are defined on a Euclidean spacetime lattice with spacing

by the overlap Dirac operator

D_ov=

	1a
	\left(\left(1+am\right)

1+\left(1-am\right)\gamma₅sign[\gamma₅A]\right)

where

is the ″kernel″ Dirac operator obeying

\gamma₅A=

	\dagger\gamma
A
	5

, i.e.

\gamma₅

-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations.^[7] A common choice for the kernel is

A=aD-1(1+s)

where

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_ov

.^[8]

Near

pa=0

the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)

D_ov=m+i{p/}

	1
	1+s

+l{O}(a)

whereas the unphysical doublers near

pa=\pi

are suppressed by a high mass

D_ov=

1a+m+i{p/}	1
	1-s

+l{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]

Notes and References

Exactly massless quarks on the lattice . 417 . 0370-2693 . 10.1016/s0370-2693(97)01368-3 . 1–2 . Physics Letters B . Elsevier BV . Neuberger, H. . 1998 . 141–144. hep-lat/9707022 . 1998PhLB..417..141N . 119372020 .
Overlap and domainwall fermions: what is the price of chirality? . Nuclear Physics B - Proceedings Supplements . 106-107 . 191–192 . 2002 . 0920-5632 . 10.1016/S0920-5632(01)01660-7 . Jansen, K.. hep-lat/0111062 . 2002NuPhS.106..191J . 2547180 .
An introduction to chiral symmetry on the lattice . 53 . 0146-6410 . 10.1016/j.ppnp.2004.05.003 . 2 . Progress in Particle and Nuclear Physics . Elsevier BV . Chandrasekharan, S. . 2004 . 373–418 . hep-lat/0405024 . 2004PrPNP..53..373C . 17473067 .
Going chiral: twisted mass versus overlap fermions . Computer Physics Communications . 169 . 1 . 362–364 . 2005 . 0010-4655 . 10.1016/j.cpc.2005.03.080 . Jansen, K.. 2005CoPhC.169..362J . subscription .
Book: Cambridge. Cambridge Lecture Notes in Physics. Introduction to Quantum Fields on a Lattice. 10.1017/CBO9780511583971. 9780511583971. Cambridge University Press. Smit, J.. 2002. 8 Chiral symmetry. 211–212. 20.500.12657/64022 . 116214756.
Book: FLAG Working Group; Aoki, S.. etal. Review of Lattice Results Concerning Low-Energy Particle Physics. 1310.8555. 10.1140/epjc/s10052-014-2890-7. Eur. Phys. J. C. 74. 116–117. 2014. A.1 Lattice actions. 9. 25972762. 4410391.
Algorithms for Dynamical Fermions . Kennedy, A.D. . 2012 . hep-lat/0607038.
Book: Gattringer. C.. Lang. C.B.. 2009. Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. 10.1007/978-3-642-01850-3. Springer. 7 Chiral symmetry on the lattice. 177–182. 978-3642018497.
Vig . Réka Á. . Kovács . Tamás G. . 2020-05-26 . Localization with overlap fermions . Physical Review D . en . 101 . 9 . 10.1103/PhysRevD.101.094511 . 2470-0010. 2001.06872 .