  We introduce a topological quantum number -- coined dynamical topological
order parameter (DTOP) -- that is dynamically defined in the real-time
evolution of a quantum many-body system and represented by a momentum space
winding number of the Pancharatnam geometric phase. Our construction goes
conceptually beyond the standard notion of topological invariants
characterizing the wave-function of a system, which are constants of motion
under coherent time evolution. In particular, we show that the DTOP can change
its integer value at discrete times where so called dynamical quantum phase
transitions occur, thus serving as a dynamical analog of an order parameter.
Interestingly, studying quantum quenches in one-dimensional two-banded
Bogoliubov de Gennes models, we find that the DTOP is capable of resolving if
the topology of the system Hamiltonian has changed over the quench.
Furthermore, we investigate the relation of the DTOP to the dynamics of the
string order parameter that characterizes the topology of such systems in
thermal equilibrium.
