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.

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