Quench dynamics of localized zero energy states in graphene

In an external magnetic field, graphene forms Landau levels with an 'anomalous' level spacing due to the low energy linear dispersion. This creates a significant difference between the Landau level splitting and Zeeman splitting, allowing for nontrivial spin dynamics within a Landau level. It is expected that the ground state of the half-filled zeroth Landau level supports Skyrmions as low energy charged excitations, which can in principle exhibit collective dynamics under an external time dependent electromagnetic field via direct coupling to the Skyrmion (topological) charge density. In order to understand the dynamics, we study the related problem of massive Dirac fermions in two dimensions after a quench from an initial state containing localized zero-modes (vortices, in particular) into a homogeneous system with a Haldane mass. The Haldane term gives the leading order effect of a high-frequency drive. The one dimensional version of the problem with mass defects also leads to localized zero modes, and fractional charge. We follow the time evolution of various order parameters in these models and look for dynamical signatures of these zero modes.