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

m are defined on a Euclidean spacetime lattice with spacing a by the overlap Dirac operatorD _{ov} = (1 a)/(\left(\left( 1+ am\right)) 1 + \left( 1- am\right)\gamma _{5} 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.^[7] A common choice for the kernel isA = aD - 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 _{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 massD _{ov} = (1 a + 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.

Notes and References

1. 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 .
2. 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 .
3. 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 .
4. 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 .
5. 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.
6. 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.
7. Algorithms for Dynamical Fermions . Kennedy, A.D. . 2012 . hep-lat/0607038.
8. 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.