Quench dynamics of Hopf insulators

Hopf insulators are exotic topological states of matter outside the standard ten-fold 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 ν 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 ν relates the pre- and post-quench Hopf invariants through ν=\(mathcal{L}-mathcal{L}₀\)bmod 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 non-trivial dynamical topology is further attributed to the emergence of π-defects in the phase band of the loop unitary. These π-defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.