Local evolutions of nodal points in two-dimensional systems with chiral symmetry

Two-dimensional systems with chiral symmetry allow stable discrete band crossings (nodal points) in Brillouin zones. Here we study the local evolutions of these nodal points under chiral symmetry preserving perturbations. We find that these evolutions can be classified by different types of local k·p models around the nodal points. Several concrete examples are calculated to illustrate our results.