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.^[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 ${\displaystyle m}$ 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]

References

1. Neuberger, H. (1998). "Exactly massless quarks on the lattice". Physics Letters B. 417 (1–2). Elsevier BV: 141–144. arXiv:hep-lat/9707022. Bibcode:1998PhLB..417..141N. doi:10.1016/s0370-2693(97)01368-3. ISSN 0370-2693. S2CID 119372020.
2. Jansen, K. (2002). "Overlap and domainwall fermions: what is the price of chirality?". Nuclear Physics B - Proceedings Supplements. 106–107: 191–192. arXiv:hep-lat/0111062. Bibcode:2002NuPhS.106..191J. doi:10.1016/S0920-5632(01)01660-7. ISSN 0920-5632. S2CID 2547180.
3. Chandrasekharan, S. (2004). "An introduction to chiral symmetry on the lattice". Progress in Particle and Nuclear Physics. 53 (2). Elsevier BV: 373–418. arXiv:hep-lat/0405024. Bibcode:2004PrPNP..53..373C. doi:10.1016/j.ppnp.2004.05.003. ISSN 0146-6410. S2CID 17473067.
4. Jansen, K. (2005). "Going chiral: twisted mass versus overlap fermions". Computer Physics Communications. 169 (1): 362–364. Bibcode:2005CoPhC.169..362J. doi:10.1016/j.cpc.2005.03.080. ISSN 0010-4655.
5. Smit, J. (2002). "8 Chiral symmetry". Introduction to Quantum Fields on a Lattice. Cambridge Lecture Notes in Physics. Cambridge: Cambridge University Press. pp. 211–212. doi:10.1017/CBO9780511583971. hdl:20.500.12657/64022. ISBN 978-0-511-58397-1. S2CID 116214756.
6. FLAG Working Group; Aoki, S.; et al. (2014). "A.1 Lattice actions". Review of Lattice Results Concerning Low-Energy Particle Physics. Eur. Phys. J. C. Vol. 74. pp. 116–117. arXiv:1310.8555. doi:10.1140/epjc/s10052-014-2890-7. PMC 4410391. PMID 25972762.
7. Kennedy, A.D. (2012). "Algorithms for Dynamical Fermions". arXiv:hep-lat/0607038. {{cite journal}}: Cite journal requires |journal= (help)
8. Gattringer, C.; Lang, C.B. (2009). "7 Chiral symmetry on the lattice". Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. Springer. pp. 177–182. doi:10.1007/978-3-642-01850-3. ISBN 978-3-642-01849-7.
9. Vig, Réka Á.; Kovács, Tamás G. (2020-05-26). "Localization with overlap fermions". Physical Review D. 101 (9) 094511. arXiv:2001.06872. Bibcode:2020PhRvD.101i4511V. doi:10.1103/PhysRevD.101.094511. ISSN 2470-0010.