Wilson fermion

Last updated July 16, 2024

In lattice field theory, Wilson fermions are a fermion discretization that allows to avoid the fermion doubling problem proposed by Kenneth Wilson in 1974.^[1] They are widely used, for instance in lattice QCD calculations.^[2]^[3]^[4]^[5]

An additional so-called Wilson term

S_{W}=-a^{d+1}\sum _{x,\mu }{\frac {i}{2a^{2}}}\left({\bar {\psi }}_{x}\psi _{x+{\hat {\mu }}}+{\bar {\psi }}_{x+{\hat {\mu }}}\psi _{x}-2{\bar {\psi }}_{x}\psi _{x}\right)

is introduced supplementing the naively discretized Dirac action in $d$ -dimensional Euclidean spacetime with lattice spacing $a$ , Dirac fields $\psi _{x}$ at every lattice point $x$ , and the vectors ${\hat {\mu }}$ being unit vectors in the $\mu$ direction. The inverse free fermion propagator in momentum space now reads^[6]

D(p)=m+{\frac {i}{a}}\sum _{\mu }\gamma _{\mu }\sin \left(p_{\mu }a\right)+{\frac {1}{a}}\sum _{\mu }\left(1-\cos \left(p_{\mu }a\right)\right)\,

where the last addend corresponds to the Wilson term again. It modifies the mass $m$ of the doublers to

m+{\frac {2l}{a}}\,

where $l$ is the number of momentum components with $p_{\mu }=\pi /a$ . In the continuum limit $a\rightarrow 0$ the doublers become very heavy and decouple from the theory.

Wilson fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry since the Wilson term does not anti-commute with $\gamma _{5}$ .

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References

↑ Wilson, K.G. (1974). "Confinement of quarks". Phys. Rev. D. 10 (8). American Physical Society: 2445-2459. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.
↑ Rothe, Heinz J. (2005). "4 Fermions on the lattice". Lattice Gauge Theories: An Introduction. World Scientific Lecture Notes in Physics (3 ed.). World Scientific Publishing Company. pp. 56–57. ISBN 978-9814365857.
↑ Smit, J. (2002). "6 Fermions on the lattice". Introduction to Quantum Fields on a Lattice. Cambridge Lecture Notes in Physics. Cambridge: Cambridge University Press. pp. 156–160. doi:10.1017/CBO9780511583971. hdl:20.500.12657/64022. ISBN 9780511583971.
↑ Montvay, I.; Münster, G. (1994). "4 Fermion fields". Quantum Fields on a Lattice. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. pp. 221–224. doi:10.1017/CBO9780511470783. ISBN 9780511470783. S2CID 118339104.{{cite book}}: CS1 maint: multiple names: authors list (link)
↑ 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. 113–115. arXiv: 1310.8555 . doi:10.1140/epjc/s10052-014-2890-7. PMC 4410391 . PMID 25972762.{{cite book}}: CS1 maint: multiple names: authors list (link)
↑ Gattringer, C.; Lang, C.B. (2009). "5 Fermions on the lattice". Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. Springer. pp. 112–114. doi:10.1007/978-3-642-01850-3. ISBN 978-3642018497.

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[1] Wilson, K.G. (1974). "Confinement of quarks". Phys. Rev. D. 10 (8). American Physical Society: 2445-2459. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.

[2] Rothe, Heinz J. (2005). "4 Fermions on the lattice". Lattice Gauge Theories: An Introduction. World Scientific Lecture Notes in Physics (3 ed.). World Scientific Publishing Company. pp. 56–57. ISBN 978-9814365857.

[3] Smit, J. (2002). "6 Fermions on the lattice". Introduction to Quantum Fields on a Lattice. Cambridge Lecture Notes in Physics. Cambridge: Cambridge University Press. pp. 156–160. doi:10.1017/CBO9780511583971. hdl:20.500.12657/64022. ISBN 9780511583971.

[4] Montvay, I.; Münster, G. (1994). "4 Fermion fields". Quantum Fields on a Lattice. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. pp. 221–224. doi:10.1017/CBO9780511470783. ISBN 9780511470783. S2CID 118339104.{{cite book}}: CS1 maint: multiple names: authors list (link)

[5] 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. 113–115. arXiv: 1310.8555 . doi:10.1140/epjc/s10052-014-2890-7. PMC 4410391 . PMID 25972762.{{cite book}}: CS1 maint: multiple names: authors list (link)

[6] Gattringer, C.; Lang, C.B. (2009). "5 Fermions on the lattice". Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. Springer. pp. 112–114. doi:10.1007/978-3-642-01850-3. ISBN 978-3642018497.

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