Physics: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 [math]\displaystyle{ m }[/math] are defined on a Euclidean spacetime lattice with spacing [math]\displaystyle{ a }[/math] by the overlap Dirac operator
- [math]\displaystyle{ D_{\text{ov}} = \frac1a \left(\left(1+am\right) \mathbf{1} + \left(1-am\right)\gamma_5 \mathrm{sign}[\gamma_5 A]\right)\, }[/math]
where [math]\displaystyle{ A }[/math] is the ″kernel″ Dirac operator obeying [math]\displaystyle{ \gamma_5 A = A^\dagger\gamma_5 }[/math], i.e. [math]\displaystyle{ A }[/math] is [math]\displaystyle{ \gamma_5 }[/math]-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations.[7] A common choice for the kernel is
- [math]\displaystyle{ A = aD - \mathbf 1(1+s)\, }[/math]
where [math]\displaystyle{ D }[/math] is the massless Dirac operator and [math]\displaystyle{ s\in\left(-1,1\right) }[/math] is a free parameter that can be tuned to optimise locality of [math]\displaystyle{ D_\text{ov} }[/math].[8]
Near [math]\displaystyle{ pa=0 }[/math] the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)
- [math]\displaystyle{ D_\text{ov} = m+i\, {p\!\!\!/}\frac{1}{1+s}+\mathcal{O}(a)\, }[/math]
whereas the unphysical doublers near [math]\displaystyle{ pa=\pi }[/math] are suppressed by a high mass
- [math]\displaystyle{ D_\text{ov} = \frac1a+m+i\,{p\!\!\!/}\frac{1}{1-s}+\mathcal{O}(a) }[/math]
and decouple.
Overlap fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry (obeying the Ginsparg–Wilson equation) and locality.[citation needed]
References
- ↑ Neuberger, H. (1998). "Exactly massless quarks on the lattice". Physics Letters B (Elsevier BV) 417 (1–2): 141–144. doi:10.1016/s0370-2693(97)01368-3. ISSN 0370-2693. Bibcode: 1998PhLB..417..141N. http://dx.doi.org/10.1016/S0370-2693(97)01368-3.
- ↑ Jansen, K. (2002). "Overlap and domainwall fermions: what is the price of chirality?". Nuclear Physics B - Proceedings Supplements 106-107: 191–192. doi:10.1016/S0920-5632(01)01660-7. ISSN 0920-5632. Bibcode: 2002NuPhS.106..191J. https://www.sciencedirect.com/science/article/pii/S0920563201016607.
- ↑ Chandrasekharan, S. (2004). "An introduction to chiral symmetry on the lattice". Progress in Particle and Nuclear Physics (Elsevier BV) 53 (2): 373–418. doi:10.1016/j.ppnp.2004.05.003. ISSN 0146-6410. Bibcode: 2004PrPNP..53..373C. http://dx.doi.org/10.1016/j.ppnp.2004.05.003.
- ↑ Jansen, K. (2005). "Going chiral: twisted mass versus overlap fermions". Computer Physics Communications 169 (1): 362–364. doi:10.1016/j.cpc.2005.03.080. ISSN 0010-4655. Bibcode: 2005CoPhC.169..362J. https://www.sciencedirect.com/science/article/pii/S0010465505001773.
- ↑ 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. ISBN 9780511583971. https://library.oapen.org/handle/20.500.12657/64022.
- ↑ FLAG Working Group; Aoki, S. (2014). "A.1 Lattice actions". Review of Lattice Results Concerning Low-Energy Particle Physics. Eur. Phys. J. C. 74. pp. 116–117. doi:10.1140/epjc/s10052-014-2890-7.
- ↑ Kennedy, A.D. (2012). Algorithms for Dynamical Fermions.
- ↑ 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.
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