Fermion doubling
Encyclopedia
The fermion doubling problem is a problem that is encountered when naively trying to put fermionic field
s 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 fermion
s.
Dirac fermion
in d dimensions,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. of mass
m, and in the continuum (i.e. without discretization) is commonly given as
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:
where a is the lattice spacing and
is the vector of length a in the μ direction. When one computes the inverse fermion propagator in momentum space, one readily finds:
Due to the finite lattice spacing the momenta pμ have to be inside the Brillouin zone
, which is typically taken to be the interval [−/a,+/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 /a), one finds the same continuum form again, although the sign in front of the gamma matrix can change.Due to these changes in sign, the chiral anomaly
cancels exactly, which is not in agreement with phenomenology. This means that, when one of the components of the momentum is near /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).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.
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:
Fermionic field
In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
s on a lattice
Lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by...
. 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 fermion
Staggered fermion
Staggered fermion is a technical subtlety that arises when fermionic fields are included in lattice gauge theory. When one does so, many new unphysical fermionic excitations corresponding to alternating fermionic fields occur in the spectrum. This is known as the fermion doubling problem...
s.
Mathematical overview
The action of a freeFree field
In classical physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution for a given initial condition....
Dirac fermion
Dirac fermion
In particle physics, a Dirac fermion is a fermion which is not its own anti-particle. It is named for Paul Dirac. All fermions in the standard model, except possibly neutrinos, are Dirac fermions...
in d dimensions,As the lattice discretization is always defined in Euclidean spacetime, we will suppose the suitable Wick rotation
Wick rotation
In physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
has been performed. Herefore, no difference will be made between covariant and cotravariant indices. of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
m, and in the continuum (i.e. without discretization) is commonly given as
Here, the Feynman slash notation
Feynman slash notation
In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient 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
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
. The resulting action is now:
where a is the lattice spacing and
is the vector of length a in the μ direction. When one computes the inverse fermion propagator in momentum space, one readily finds:Due to the finite lattice spacing the momenta pμ have to be inside the Brillouin zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
, which is typically taken to be the interval [−/a,+/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 /a), one finds the same continuum form again, although the sign in front of the gamma matrix can change.Due to these changes in sign, the chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
cancels exactly, which is not in agreement with phenomenology. This means that, when one of the components of the momentum is near /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).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.
The Nielsen–Ninomiya theorem
NielsenHolger Bech Nielsen
Holger Bech Nielsen is a Danish theoretical physicist, professor at the Niels Bohr Institute, at the University of Copenhagen, where he started studying physics in 1961....
and Ninomiya proved a theorem stating that a local, real, free fermion lattice action, having chiral
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
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 fermionStaggered fermionStaggered fermion is a technical subtlety that arises when fermionic fields are included in lattice gauge theory. When one does so, many new unphysical fermionic excitations corresponding to alternating fermionic fields occur in the spectrum. This is known as the fermion doubling problem...
s - Twisted mass fermions
- Ginsparg–Wilson fermions
- Domain wall fermions
- Overlap fermions
See also
- Staggered fermionStaggered fermionStaggered fermion is a technical subtlety that arises when fermionic fields are included in lattice gauge theory. When one does so, many new unphysical fermionic excitations corresponding to alternating fermionic fields occur in the spectrum. This is known as the fermion doubling problem...
s: a way to reduce the number of doublers - Acoustic and optical phonons: a similar phenomenon in solid state crystals