Niemeier lattice: Difference between revisions

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The '''fermion doubling problem''' is a problem that is encountered when naively trying to put [[fermionic field]]s on a [[lattice field theory|lattice]]. It consists in the appearance of spurious states, such that one ends up having 2<sup>''d''</sup> fermionic particles (with ''d'' the number of discretized dimensions) for each original fermion. In order to solve this problem, several strategies are in use, such as [[Wilson fermion]]s and [[staggered fermion]]s.
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==Mathematical overview==
The action of a [[free field|free]] [[Dirac fermion]] in ''d'' dimensions,<ref group="note">As the lattice discretization is always defined in Euclidean spacetime, we will suppose the suitable [[Wick rotation]] has been performed. Herefore, no difference will be made between covariant and cotravariant indices.</ref> of [[mass]] ''m'', and in the continuum (i.e. without discretization) is commonly given as
:<math> S = \int d^dx \; \bar\psi (\partial\!\!\!/ + m) \psi \;.</math>
Here, the [[Feynman slash notation]] was used to write
:<math> \gamma_\mu a^{\mu} = a\!\!\!/ </math>
where γ<sub>μ</sub> are the [[gamma matrices]]. When this action is discretized on a cubic lattice, the fermion field ψ(''x'') is replaced with a discretized version ψ<sub>''x''</sub>, where ''x'' now denotes the lattice site. The derivative is replaced by the [[finite difference]]. The resulting action is now:<ref name=wiese>{{Cite journal | last1 = Chandrasekharan | last2 = Wiese | title = An Introduction to chiral symmetry on the lattice | journal = Prog. Part. Nucl. Phys. | volume = 53 | pages = 373 | date = 2004 | url = http://www.sciencedirect.com/science/article/pii/S0146641004000948 | doi = 10.1016/j.ppnp.2004.05.003|arxiv = hep-lat/0405024 |bibcode = 2004PrPNP..53..373C }}</ref>
:<math> S = a^d \sum_{x,\mu} \frac1{2a} (\bar\psi_x\gamma_\mu\psi_{x+\hat\mu} - \bar\psi_{x+\hat\mu}\gamma_\mu\psi_x) + a^d \sum_x m\bar\psi_x\psi_x \;,</math>
where ''a'' is the lattice spacing and <math>\hat\mu</math> is the vector of length ''a'' in the μ direction. When one computes the inverse fermion propagator in [[momentum space]], one readily finds:<ref name="wiese" />
:<math> S^{-1}(p) = m + \frac ia \sum_\mu \gamma_\mu \sin(p_\mu a) \;. </math>
Due to the finite lattice spacing the momenta ''p''<sub>μ</sub> have to be inside the [[Brillouin zone]], which is typically taken to be the interval [−{{Pi}}/''a'',+{{Pi}}/''a''].
 
When simply taking the limit ''a'' → 0 in the above inverse propagator, one recovers the correct continuum result. However, when instead expanding this expression around a value of ''p''<sub>μ</sub> where one or more of the components are at the corners of the Brillouin zone (i.e. equal to {{Pi}}/''a''), one finds the same continuum form again, although the sign in front of the gamma matrix can change.<ref>{{cite arxiv | last = Gupta | eprint = hep-lat/9807028 | title = Introduction to lattice QCD | year = 1998}}</ref><ref group="note">Due to these changes in sign, the [[chiral anomaly]] cancels exactly, which is not in agreement with phenomenology.</ref> This means that, when one of the components of the momentum is near {{Pi}}/''a'', the discretized fermion field will again behave like a continuum fermion. This can happen with all ''d'' components of the momentum, leading to —with the original fermion with momentum near the origin included— 2<sup>''d''</sup> different "tastes" (in analogy to [[flavor (particle physics)|flavor]]).<ref group="note">As the action of scalars contains second derivatives, a similar procedure in this case would lead to a quadratic inverse propagator, which does not have these doublers.</ref>
 
==The Nielsen–Ninomiya theorem==
[[Holger Bech Nielsen|Nielsen]] and Ninomiya proved a theorem<ref>{{Cite journal | last1 = Nielsen | last2 = Ninomiya | authorlink1 = Holger Bech Nielsen | title = Absence of Neutrinos on a Lattice | journal = Nucl. Phys. B | volume = 185 | year = 1981 | pages = 20 | doi = 10.1016/0550-3213(81)90361-8|bibcode = 1981NuPhB.185...20N }}<br/>{{Cite journal | last1 = Nielsen | last2 = Ninomiya | authorlink1 = Holger Bech Nielsen | title = No go theorem for regularizing chiral fermions | journal = Phys. Lett. B | volume = 105 | year = 1981 | pages = 219 | doi = 10.1016/0370-2693(81)91026-1|bibcode = 1981PhLB..105..219N }}</ref> stating that a local, real, free fermion lattice action, having [[chiral symmetry|chiral]] and translational invariance, necessarily has fermion doubling. The only way to get rid of the doublers is by violating one of the presuppositions of the theorem —for example:
* [[Wilson fermion]]s explicitly violate chiral symmetry, giving an infinitely high mass to the doublers which then decouple.
* So-called "[[perfect fermions|perfect lattice fermions]]" have a nonlocal action.
* [[Staggered fermion]]s
* [[Twisted mass fermions]]
* [[Ginsparg–Wilson fermions]]
* [[Domain wall fermions]]
* [[Overlap fermions]]
 
==See also==
* [[Staggered fermion]]s: a way to reduce the number of doublers
* [[Optical phonon#Acoustic and optical phonons|Acoustic and optical phonons]]: a similar phenomenon in solid state crystals
 
==Notes and references==
;Notes
<references group="note" />
 
;References
{{Reflist|close=1}}
 
[[Category:Lattice models]]
[[Category:Fermions]]

Latest revision as of 17:26, 28 June 2014

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