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We show that dynamical quantum phase transitions (DQPTs) in the quenchdynamics of two-dimensional topological systems can be characterized by adynamical topological invariant defined along an appropriately chosen closedcontour in momentum space. Such a dynamical topological invariant reflects thevorticity of dynamical vortices responsible for the DQPTs, and thus serves as adynamical topological order parameter in two dimensions. We demonstrate thatwhen the contour crosses topologically protected fixed points in the quenchdynamics, an intimate connection can be established between the dynamicaltopological order parameter in two dimensions and those in one dimension. Wefurther define a reduced rate function of the Loschmidt echo on the contour,which features non-analyticities at critical times and is sufficient tocharacterize DQPTs in two dimensions. We illustrate our results using theHaldane honeycomb model and the quantum anomalous Hall model as concreteexamples, both of which have been experimentally realized using cold atoms.

Dynamical topological invariants and reduced rate functions for dynamical quantum phase transitions in two dimensions