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,[ 1] [ 2] [ 3] [ 4] lattice QCD .[ 5] [ 6]
Overlap fermions with mass
m
{\displaystyle m}
Euclidean spacetime lattice with spacing
a
{\displaystyle a}
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} [\gamma _{5}A]\right)\,}
where
A
{\displaystyle A}
γ
5
A
=
A
†
γ
5
{\displaystyle \gamma _{5}A=A^{\dagger }\gamma _{5}}
A
{\displaystyle A}
γ
5
{\displaystyle \gamma _{5}}
rational approximations .[ 7]
A
=
a
D
−
1
(
1
+
s
)
{\displaystyle A=aD-\mathbf {1} (1+s)\,}
where
D
{\displaystyle D}
s
∈
(
−
1
,
1
)
{\displaystyle s\in \left(-1,1\right)}
D
ov
{\displaystyle D_{\text{ov}}}
[ 8]
Near
p
a
=
0
{\displaystyle pa=0}
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 }
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.[ 9]
References
^ 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 . ^ 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 . ^ 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 . ^ 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 . ^ Smit, J. (2002). "8 Chiral symmetry". Introduction to Quantum Fields on a Lattice 211– 212. doi :10.1017/CBO9780511583971 . hdl :20.500.12657/64022 . ISBN 9780511583971 S2CID 116214756 . ^ 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 )^ Kennedy, A.D. (2012). "Algorithms for Dynamical Fermions". arXiv :hep-lat/0607038 ^ 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 ^ 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 . {{cite journal }}: CS1 maint: article number as page number (link )
![{\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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3d62da533fc04fb93264ecd6463d3f87b1cc8e)
