Overlap fermion

{{technical|date=September 2023}}

{{Short description|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,{{cite journal

|title=Exactly massless quarks on the lattice

|volume=417

|issn=0370-2693

|url=http://dx.doi.org/10.1016/S0370-2693(97)01368-3

|doi=10.1016/s0370-2693(97)01368-3

|number=1–2

|journal=Physics Letters B

|publisher=Elsevier BV

|author=Neuberger, H.

|year=1998

|pages=141–144|arxiv=hep-lat/9707022

|bibcode=1998PhLB..417..141N

|s2cid=119372020

}} they were quickly taken up for a variety of numerical simulations.{{cite journal

|title = Overlap and domainwall fermions: what is the price of chirality?

|journal = Nuclear Physics B - Proceedings Supplements

|volume = 106-107

|pages = 191–192

|year = 2002

|issn = 0920-5632

|doi = 10.1016/S0920-5632(01)01660-7

|url = https://www.sciencedirect.com/science/article/pii/S0920563201016607

|author = Jansen, K.|arxiv = hep-lat/0111062

|bibcode = 2002NuPhS.106..191J

|s2cid = 2547180

}}{{cite journal

|title=An introduction to chiral symmetry on the lattice

|volume=53

|issn=0146-6410

|url=http://dx.doi.org/10.1016/j.ppnp.2004.05.003

|doi=10.1016/j.ppnp.2004.05.003

|number=2

|journal=Progress in Particle and Nuclear Physics

|publisher=Elsevier BV

|author=Chandrasekharan, S.

|year=2004

|pages=373–418 |arxiv=hep-lat/0405024

|bibcode=2004PrPNP..53..373C

|s2cid=17473067

}}{{cite journal

|title = Going chiral: twisted mass versus overlap fermions

|journal = Computer Physics Communications

|volume = 169

|number = 1

|pages = 362–364

|year = 2005

|issn = 0010-4655

|doi = 10.1016/j.cpc.2005.03.080

|url = https://www.sciencedirect.com/science/article/pii/S0010465505001773

|author = Jansen, K.|bibcode = 2005CoPhC.169..362J

|url-access = subscription

}} By now overlap fermions are well established and regularly used in non-perturbative fermion simulations, for instance in lattice QCD.{{cite book|place=Cambridge|series=Cambridge Lecture Notes in Physics|title=Introduction to Quantum Fields on a Lattice|doi=10.1017/CBO9780511583971|isbn = 9780511583971|publisher=Cambridge University Press|author=Smit, J.|year=2002|chapter = 8 Chiral symmetry|pages = 211–212|hdl=20.500.12657/64022 |s2cid=116214756|url=https://library.oapen.org/handle/20.500.12657/64022}}{{cite book|author=FLAG Working Group; Aoki, S.|display-authors=etal|title = Review of Lattice Results Concerning Low-Energy Particle Physics

|arxiv=1310.8555|doi = 10.1140/epjc/s10052-014-2890-7|series = Eur. Phys. J. C|volume = 74|pages = 116–117|date = 2014|chapter=A.1 Lattice actions|issue=9|pmid=25972762|pmc=4410391}}

Overlap fermions with mass $m$ are defined on a Euclidean spacetime lattice with spacing $a$ by the overlap Dirac operator

:$$

D_{\text{ov}} = \frac1a \left(\left(1+am\right) \mathbf{1} + \left(1-am\right)\gamma_5 \mathrm{sign}[\gamma_5 A]\right)\,

where $A$ is the ″kernel″ Dirac operator obeying $\backslash gamma\_5\; A\; =\; A^\backslash dagger\backslash gamma\_5$, i.e. $A$ is $\backslash gamma\_5$-hermitian. The sign-function usually has to be calculated numerically, e.g. by rational approximations.{{cite journal

|title=Algorithms for Dynamical Fermions

|author= Kennedy, A.D.

|date=2012

|arxiv=hep-lat/0607038}} A common choice for the kernel is

:$$

A = aD - \mathbf 1(1+s)\,

where $D$ is the massless Dirac operator and $s\backslash in\backslash left(-1,1\backslash right)$ is a free parameter that can be tuned to optimise locality of $D\_\backslash text\{ov\}$.{{cite book|last1=Gattringer|first1=C.|last2=Lang|first2=C.B.|date=2009|title=Quantum Chromodynamics on the Lattice: An Introductory Presentation|series=Lecture Notes in Physics 788|url=|doi=10.1007/978-3-642-01850-3|location=|publisher=Springer|chapter=7 Chiral symmetry on the lattice|pages=177–182|isbn=978-3642018497}}

Near $pa=0$ the overlap Dirac operator recovers the correct continuum form (using the Feynman slash notation)

:$$

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

whereas the unphysical doublers near $pa=\backslash pi$ are suppressed by a high mass

:$$

D_\text{ov} = \frac1a+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.{{Cite journal |last1=Vig |first1=Réka Á. |last2=Kovács |first2=Tamás G. |date=2020-05-26 |title=Localization with overlap fermions |url=https://link.aps.org/doi/10.1103/PhysRevD.101.094511 |journal=Physical Review D |language=en |volume=101 |issue=9 |page=094511 |doi=10.1103/PhysRevD.101.094511 |issn=2470-0010|arxiv=2001.06872 |bibcode=2020PhRvD.101i4511V }}