Fermion doubling

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 2^d 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.

The action of a free Dirac fermion in d dimensions, 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 γ_μ 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_x, 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 ${\displaystyle {\hat {\mu }}}$ 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 p_μ have to be inside the (first) Brillouin zone, which is typically taken to be the interval [−π/a,+π/a].

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