Topological quantum quench dynamics carrying arbitrary Hopf and second-Chern numbers

A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group $\pi _{3}(S^{2})=\mathbb{Z}$. When we quench a system from an insulator with the Chern number $C_{i}\in \pi _{2}(S^{2})=\mathbb{Z}$ to another insulator with the Chern number $C_{f}$, the preimage of the Hamiltonian vector forms links having the Hopf number $C_{f}-C_{i}$. We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second-Chern number $N\in \pi _{4}(S^{4})=\mathbb{Z}$, which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N\in \pi _{3}(S^{3})=\mathbb{Z}$.