Using the two dimensional Gross-Pitaevskii (a.k.a. nonlinear Schroedinger) equation as a concrete example, we explain the concepts of Hamiltonian time crystals and anyons. Specifically, we first review the phenomenology of the GP equation, the way it describes vortices in cold atom Bose-Einstein condensates. We then expand the conventional point of view, and show that the minimum energy states are time crystalline vortices. We proceed to show that their exchange statistics has anyon structure. We then identify a Kosterlitz-Thouless topological transition and explain how it can be analyzed using the Poincare index formula.
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