LAUSR

Hopf insulators are exotic topological states of matter outside the standard tenfold-way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a ${\mathbb{Z}}_{2}$ invariant $\ensuremath{\nu}$ which reveals a rich interplay between quantum quench and static band topology. We construct the ${\mathbb{Z}}_{2}$ topological invariant using the loop unitary operator and prove that $\ensuremath{\nu}$ relates the pre- and postquench Hopf invariants through $\ensuremath{\nu}=(\mathcal{L}\ensuremath{-}{\mathcal{L}}_{0})\phantom{\rule{0.28em}{0ex}}mod\phantom{\rule{0.28em}{0ex}}2$. The ${\mathbb{Z}}_{2}$ nature of the dynamical invariant is in sharp contrast to the $\mathbb{Z}$ invariant for the quench dynamics of Chern insulators in two dimensions. The nontrivial dynamical topology is further attributed to the emergence of $\ensuremath{\pi}$ defects in the phase band of the loop unitary. These $\ensuremath{\pi}$ defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.