Dynamical formation of a magnetic polaron in a two-dimensional quantum antiferromagnet

We numerically study the real-time dynamics of a single hole created in the $t-J$ model on a square lattice. Initially, the hole spreads ballistically with a velocity proportional to the hopping matrix element. At intermediate to long times, the dimensionality as well as the spin background determine the hole dynamics. A hole created in the ground state of a two dimensional quantum antiferromagnet propagates again ballistically at long times but with a velocity proportional to the spin exchange coupling, showing the formation of a magnetic polaron. We provide an intuitive explanation of this dynamics in terms of a parton construction, which leads to a good quantitative agreement with the numerical simulations. In the limit of infinite temperature and no spin exchange couplings, the dynamics can be approximated by a quantum random walk on the Bethe lattice. Adding Ising interactions corresponds to an effective disordered potential, which can dramatically slow down the hole propagation, consistent with subdiffusive dynamics.