The fermion doubling problem is a problem that is encountered when naively trying to put fermionic fields on a lattice. It consists in the appearance of spurious states, such that one ends up having 2d 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 fermions and staggered fermions.
Here, the Feynman slash notation was used to write
where γμ are the gamma matrices. When this action is discretized on a cubic lattice, the fermion field ψ(x) is replaced with a discretized version ψx, where x now denotes the lattice site. The derivative is replaced by the finite difference. The resulting action is now:
Due to the finite lattice spacing the momenta pμ have to be inside the Brillouin zone, which is typically taken to be the interval [−π/a,+π/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 pμ where one or more of the components are at the corners of the Brillouin zone (i.e. equal to π/a), one finds the same continuum form again, although the sign in front of the gamma matrix can change.[note 2] This means that, when one of the components of the momentum is near π/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— 2d different "tastes" (in analogy to flavor).[note 3]
The Nielsen–Ninomiya theorem
Nielsen and Ninomiya proved a theorem stating that a local, real, free fermion lattice action, having 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 fermions explicitly violate chiral symmetry, giving an infinitely high mass to the doublers which then decouple.
- So-called "perfect lattice fermions" have a nonlocal action.
- Staggered fermions
- Twisted mass fermions
- Ginsparg–Wilson fermions
- Domain wall fermions
- Overlap fermions
- Staggered fermions: a way to reduce the number of doublers
- Acoustic and optical phonons: a similar phenomenon in solid state crystals
Notes and references
- 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.
- Due to these changes in sign, the chiral anomaly cancels exactly, which is not in agreement with phenomenology.
- 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.
- Chandrasekharan; Wiese (2004). "An Introduction to chiral symmetry on the lattice". Prog. Part. Nucl. Phys. 53: 373. arXiv:hep-lat/0405024. Bibcode:2004PrPNP..53..373C. doi:10.1016/j.ppnp.2004.05.003.
- Gupta (1998). "Introduction to lattice QCD". arXiv:hep-lat/9807028.
- Nielsen; Ninomiya (1981). "Absence of Neutrinos on a Lattice". Nucl. Phys. B 185: 20. Bibcode:1981NuPhB.185...20N. doi:10.1016/0550-3213(81)90361-8.
Nielsen; Ninomiya (1981). "No go theorem for regularizing chiral fermions". Phys. Lett. B 105: 219. Bibcode:1981PhLB..105..219N. doi:10.1016/0370-2693(81)91026-1.