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 $\nu$ 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 $\nu$ relates the pre- and postquench Hopf invariants through $\nu =(\mathcal{L}-{\mathcal{L}}_0)mod2$. 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 $\pi$ defects in the phase band of the loop unitary. These $\pi$ defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.

• Received 20 December 2019
• Accepted 1 April 2020

DOI:https://doi.org/10.1103/PhysRevB.101.155131