  Dynamic quantum phase transitions (DQPT) following quantum quenches exhibit
singular behavior of the overlap between the initial and evolved states. Here
we present a formalism to incorporate a geometric phase into quench dynamics of
mixed quantum states, a process named the Uhlmann quench, based on the Uhlmann
parallel transport. To overcome the incompatibility between the Uhlmann
parallel-transport condition and Hamiltonian dynamics, we formulate the
evolution of purification of the density matrix in a form which not only
respects the dynamics according to the density matrix but also follows the
Uhlmann parallel-transport condition to generate a geometric phase after a
quantum quench. For cyclic processes exemplified by a spin-1/2 system,
geometric DQPTs (GDQPTs) can emerge with both singular behavior in the dynamic
analogue of the free energy and jumps of the geometric phase. Moreover, the
Uhlmann phase reflecting the holonomy is generated at the end of each cycle.
The Uhlmann quench thus paves the way for investigating the interplay between
quantum dynamics and geometric processes in mixed states.
