In lattice field theory, staggered fermions (also known as Kogut–Susskind fermions) are a fermion discretization that reduces the number of fermion doublers from sixteen to four. 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. Staggered fermions were first formulated by John Kogut and Leonard Susskind in 1975 and were later found to be equivalent to the discretized version of the Dirac–Kähler fermion.
Constructing staggered fermions
Single-component basis The naively discretized Dirac action in Euclidean spacetime with lattice spacing a {\displaystyle a} and Dirac fields ψ n {\displaystyle \psi _{n}} at every lattice point, indexed by n = ( n 1 , n 2 , n 3 , n 4 ) {\displaystyle n=(n_{1},n_{2},n_{3},n_{4})} , takes the form
S = a 4 ∑ n ∈ Λ ψ ¯ n ( ∑ μ = 1 4 γ μ ψ n + μ ^ − ψ n − μ ^ 2 a + m ψ n ) . {\displaystyle S=a^{4}\sum _{n\in \Lambda }{\bar {\psi }}_{n}{\bigg (}\sum _{\mu =1}^{4}\gamma _{\mu }{\frac {\psi _{n+{\hat {\mu }}}-\psi _{n-{\hat {\mu }}}}{2a}}+m\psi _{n}{\bigg )}.}
Staggered fermions are constructed from this by performing the staggered transformation into a new basis of fields ψ n ′ {\displaystyle \psi _{n}'} defined by
ψ n = γ 1 n 1 γ 2 n 2 γ 3 n 3 γ 4 n 4 ψ n ′ . {\displaystyle \psi _{n}=\gamma _{1}^{n_{1}}\gamma _{2}^{n_{2}}\gamma _{3}^{n_{3}}\gamma _{4}^{n_{4}}\psi '_{n}.}
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. Its effect is to diagonalize the action in the spinor indices, meaning that the action ends up splitting into four distinct parts, one for each Dirac spinor component. Denoting one of those components by χ n {\displaystyle \chi _{n}} , which is Grassmann variable with no spin structure, the other three components can be dropped, yielding the single-component staggered action
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