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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 witha velocity proportional to the hopping matrix element. At intermediate to longtimes, the dimensionality as well as the spin background determine the holedynamics. A hole created in the ground state of a two dimensional quantumantiferromagnet propagates again ballistically at long times but with avelocity proportional to the spin exchange coupling, showing the formation of amagnetic polaron. We provide an intuitive explanation of this dynamics in termsof a parton construction, which leads to a good quantitative agreement with thenumerical simulations. In the limit of infinite temperature and no spinexchange couplings, the dynamics can be approximated by a quantum random walkon the Bethe lattice. Adding Ising interactions corresponds to an effectivedisordered potential, which can dramatically slow down the hole propagation,consistent with subdiffusive dynamics.

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