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Overlap fermion

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Lattice fermion discretisation

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

D ov = 1 a ( ( 1 + a m ) 1 + ( 1 − a m ) γ 5 s i g n [ γ 5 A ] ) {\displaystyle D_{\text{ov}}={\frac {1}{a}}\left(\left(1+am\right)\mathbf {1} +\left(1-am\right)\gamma _{5}\mathrm {sign} \right)\,}

where A {\displaystyle A} is the ″kernel″ Dirac operator obeying γ 5 A = A † γ 5 {\displaystyle \gamma _{5}A=A^{\dagger }\gamma _{5}} , i.e. A {\displaystyle A} is γ 5 {\displaystyle \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 = a D − 1 ( 1 + s ) {\displaystyle A=aD-\mathbf {1} (1+s)\,}

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

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

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

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

D ov = 1 a + m + i p / 1 1 − s + O ( a ) {\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.

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 9780511583971. 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.{{cite book}}: CS1 maint: multiple names: authors list (link)
  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 978-3642018497.
  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.
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