We discuss the continuum limit of discrete Dirac operators on the square lattice in $\mathbb R^2$ as the mesh size $h$ tends to zero. To this end, we propose the most natural and simplest embedding of $\ell^2(\mathbb Z_h^d)$ into $L^2(\mathbb R^d)$, which enables us to define difference operators in a subspace of step functions in $L^2(\mathbbR^d)$ and compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space $L^2(\mathbb R^2)^2$. Our main concern is to prove that the discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on $\mathbb R^2$ and allowed to be complex matrix-valued. We also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense. This is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis.
This is joint work with Karl Michael Schmidt.