Continuum limits for discrete Dirac operators on 2D square lattices

We discuss the continuum limit of discrete Dirac operators on the square lattice in $mathbb R^2$ as the mesh size tends to zero. To this end, we propose a natural and simple embedding of $ell^2(mathbb Z_h^d)$ into $L^2(mathbb R^d)$ that enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space $L^2(mathbb R^2)^2$. In particular, we prove strong resolvent convergence. Potentials are assumed to be bounded and uniformly continuous functions on $mathbb R^2$ and allowed to be complex matrix-valued.