Fermion Doubling - Mathematical Overview

Mathematical Overview

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 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 Brillouin zone, which is typically taken to be the interval .

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 p_μ 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. 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).