A survey on topological solitons in planar nonlinear Dirac models

In this text, we review tardyonic and tachyonic planar Dirac models with several cubic nonlinearities such as the Kerr, Soler and massive Thirring nonlinearities. These models have relevance in a newly discovered class of solids named topological insulators and have been shown to exhibit topological properties such as the Berry phase and Chern number. Moreover, nonlinear Dirac models have vortex solutions and edge state solitons. As understood in quantum mechanics that a Hamiltonian must yield real eigenvalues or dispersion relation, we argue that Dirac equations in the tachyonic case are valid when studied in non-vanishing backgrounds because this gets rid of the complexity in the dispersion relation.