They are one of the fastest lattice fermions when it comes to simulations and they also possess some nice features such as a remnant chiral symmetry, making them very popular in lattice QCD calculations.
[2] The naively discretized Dirac action in Euclidean spacetime with lattice spacing
defined by[3] Since Dirac matrices square to the identity, this position dependent transformation mixes the fermion spin components in a way that repeats itself every two lattice spacings.
, which is Grassmann variable with no spin structure, the other three components can be dropped, yielding the single-component staggered action where
are used to define the spin-taste basis of staggered fermions[5] The taste index
This change of basis turns the one-component action on the lattice with spacing
Since the kinetic and mass terms are diagonal in the taste indices, the action describes four degenerate Dirac fermions.
This action is very similar to the action constructed using four Wilson fermions with the only difference being in the second term tensor structure, which for Wilson fermions is spin and taste diagonal
The symmetry also protects massless fermions from gaining a mass upon renormalization.
where Wilson lines are inserted between the different lattice points within the hypercube to ensure gauge invariance.
[6] The resulting action cannot be expressed in a closed form but can be expended out in powers of the lattice spacing, leading to the usual interacting Dirac action for four fermions, together with an infinite series of irrelevant fermion bilinear operators that vanish in the continuum limit.
Staggered fermions can also be formulated in momentum space by transforming the single-component action into Fourier space and splitting up the Brillouin zone into sixteen blocks.
Shifting these to the origin yields sixteen copies of the single-component fermion whose momenta extend over half the Brillouin zone range
matrix which upon a unitary transformation and a momentum rescaling, to ensure that the momenta again range over the full
This is however only achieved at the expense of locality, where now the position-space Dirac operator connects lattice points that are arbitrarily far apart, rather than ones restricted to a hypercube.
Chiral symmetry is maintained despite the possibility of simulating a single momentum space fermion because locality was one of the assumptions of the Nielsen–Ninomiya theorem determining whether a theory experiences fermion doubling.
The main issue with simulating staggered fermions is that the different tastes mix together due to the taste-mixing term.
If there was no mixing between tastes, lattice simulations could easily untangle the different contributions from the different tastes to end up with the results for processes involving a single fermion.
Instead the taste mixing introduces discretization errors that are hard to account for.
The main method to reduce these errors is to perform Symanzik improvement, whereby irrelevant operators are added to the action with their coefficients fine-tuned to cancel discretization errors.
The main code and gauge ensembles used for staggered fermions comes from the MILC collaboration.
[9] An advantage of staggered fermions over some other lattice fermions in that the remnant chiral symmetry protects simulations from exceptional configurations, which are gauge field configurations that lead to small eigenvalues of the Dirac operator, making numerical inversion difficult.
Staggered fermions are protected from this because their Dirac operator is anti-hermitian, so its eigenvalues come in complex conjugate pairs
This ensures that the Dirac determinant is real and positive for non-zero masses.
This degeneracy is broken by taste mixing at non-zero lattice spacings
[10] If it does, then there is no reason to suppose that rooted staggered fermions are any good at describing the continuum field theory.
The universality class is generally determined by the dimensionality of the theory and by what symmetries it satisfies.
The problem with rooted staggered fermions is that they can only be described by a nonlocal action for which the universality classification no longer applies.
As nonlocality implies a violation of unitary, rooted staggered fermions are also non-physical at non-zero lattice spacings, although this is not a problem if the nonlocality vanishes in the continuum.
It remains an open question whether this is also true non-perturbatively, however theoretical arguments[11] and numerical comparisons to other lattice fermions indicate that rooted staggered fermions do belong to the correct universality class.