Quench dynamics of Hopf insulators

TL;DR

This paper explores the quench dynamics of Hopf insulators, revealing a $Z_2$ dynamical invariant that links static topology with non-equilibrium behavior, highlighting unique topological features like $ ext{pi}$-defects.

Contribution

It introduces a $Z_2$ invariant for Hopf insulator quench dynamics, connecting static Hopf invariants with dynamical properties through the loop unitary operator.

Findings

The dynamical invariant $ u$ relates pre- and post-quench Hopf invariants.

Non-trivial dynamics involve $ ext{pi}$-defects in phase bands.

$Z_2$ nature contrasts with $Z$ invariants in Chern insulators.

Abstract

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 $\nu$ 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 post-quench Hopf invariants through $\nu=(\mathcal{L}-\mathcal{L}_0)\bmod 2$. 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 $\pi$-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.