  We classify the topology of quench dynamics by homotopy groups. A relation
between the topological invariant of a post-quench order parameter and the
topological invariant of a static Hamiltonian is shown in one, two and three
dimensions. The mid-gap states in the entanglement spectrum of the post-quench
state reveal its topological nature. When a trivial quantum state under a
sudden quench to a Chern insulator, the mid-gap states in entanglement spectrum
form rings. These rings are analogous to the boundary Fermi rings in the Hopf
insulators. Finally, we show a post-quench state in 3+1 dimensions can be
characterized by the second Chern number. The number of Dirac cones in the
entanglement spectrum is equal to the second Chern number.

Mesoscale and Nanoscale Physics;Quantum Gases