Fermion doubling

In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice.

A lattice is a periodic arrangement of vertices. If we Fourier transform a lattice, the space of momenta is a torus with the shape of the fundamental domain of the reciprocal lattice called the Brillouin zone.

This means if we look at the wave solutions over a lattice, the eigenvalue of the fermion operator as a function of momentum (aka wave vector) has to be periodic.

For a free bosonic field, the action is quadratic, so the eigenvalues tends to have the form

:$lambda\; propto\; sqrt\{4sin^2(kL/2)/L^2+m^2\}$

or a similar form where m<<1/L. At scales much larger than the lattice spacing (i.e. for eigenvalues close to zero) only the momenta around k=0 dominate and we have a single species of boson.

Fermions, on the other hand, are described by first order equations. So, we might have something which goes like

:$lambda\; propto\; \{sin(kL)over\; L\}$

at least with one spatial dimension, but the higher dimensional cases are analogous. If we look at the low eigenvalue limit, we see two different regions; one about k=0 and the other about k=&pi;/L. They behave like two different kinds of particles. This is called fermion doubling and each species of fermions is called a taste (in analogy to flavor).

Fermion doubling is a generic consequence of local actions and Hamiltonians. A way to get rid of these unwanted doublers was first proposed by Wilson.

A new term, the Wilson term, is added to the fermion action and removes the doublers (or better, it makes the doublers infinitely heavy and thus unobservable). However, removing the doublers does not come for free because the additional term explicitly breaks a fundamental symmetry of QCD, the chiral symmetry.

One way of seeing why this is a generic feature is to look at the Fermi points of the model. Generically, there will be more than one Fermi point (an even number, in fact).

See also staggered fermions