In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice. See lattice for other mathematical as well as non-mathematical meanings of the term. ...

A lattice is a periodic arrangement of vertices. If you Fourier transform a lattice, the space of momenta is a torus with the shape of the fundamental domain of the reciprocal lattice. In colloquial usage, a lattice is a structure of crossed laths with open spaces left between them. ... The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... Geometry In geometry, a torus (pl. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice. ...

This means if we look at the wave solutions over a lattice, the energy (aka frequency) as a function of momentum (aka wave vector) has to be periodic. A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

For a bosonic field, the action is quadratic and so, the energy tends to have the form

or something like that where m<<1/L. At scales much larger than the lattice spacing (i.e. at low energies) 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

at least with one spatial dimension, but the higher dimensional cases are analogous. If we look at the low energy limit, we see two different regions; one about k=0 and the other about k=π/L. They behave like two different kinds of particles. This is called fermion doubling.

Fermion doubling is a generic consequence of local actions and Hamiltonians. However, the Ginsparg-Wilson action, which is nonlocal solves it (partially, by reducing the number of species). In physics, Hamiltonian has distinct but closely related meanings. ... Nonlocality in quantum mechanics, refers to the property of entangled quantum states in which both the entangled states collapse simultaneously upon measurement of one of their entangled components, regardless of the spatial separation of the two states. ...