Chiral Dirac fermions on the lattice using Geometric Discretisation

The theorem of Nielsen and Ninomiya [3] came with a topological proof of the fact that under reasonable assumptions, fermion doubling is unavoidable on the lattice. A key element of the proof was the periodicity of the Brillouin zone thus setting the approach in momentum space. In other arguments [4], [5] it was argued that doubling is already present when one starts one step back and consider the DK equation (1) and it persists after reduction to the Dirac equation on the lattice. The discretised field is then an inhomogeneous cochain taking value on points, edges and so on and it is a 16D object in four dimensions. The DK equation is also particularly well suited for counting fields through its link with the Laplacian and consequently with homology. Accordingly, the method of Rabin provided us with a position space doubling or “species doubling” and the conceptual picture was new: the failure in constructing well-defined discrete analogies to the basic operands used in (1) was effectively the way doubling survived. Specifically, the Hodge star (⋆) which is the analogue of γ5, and the various properties relating it with the DK operator did not hold simultaneously. The picture is then: i) doubling has an algebraic formulation; and ii) it might be addressed at the classical level, that is at the level of the action. Meanwhile Becher and Joos attempted to carry