From orthogonal link to phase vortex in generalized dynamical Hopf insulators

TL;DR

This paper introduces a new approach to understanding quenched 2D Chern insulators by linking phase vortex trajectories to the difference in Chern numbers, bypassing traditional Hopf invariant methods.

Contribution

It presents a novel recipe for analyzing dynamical topological properties in quenched 2D insulators, emphasizing phase vortex trajectories over Hopf invariants.

Findings

Linking number equals the difference in pre- and post-quench Chern numbers.

Phase vortex trajectories maintain standard link shapes regardless of initial topological state.

The approach is demonstrated through two concrete examples with polarity reversal at fixed points.

Abstract

In the creation of Hopf topological matters, the old paradigm is to conceive the Hopf invariant first, and then display its intuitive topology through links. Here we brush aside this effort and put forward a new recipe for unraveling the quenched two-dimensional (2D) two-band Chern insulators under a parallel quench protocol, which implies that the quench quantities with different momentum k are parallel or antiparallel to each other. We find that whether the dynamical Hopf invariant exists or not, the links in (2+1)D space always keep their standard shape even for topological initial states, and trace out the trajectories of phase vortices. The linking number is exactly equal to the difference between pre- and post-quench Chern numbers regardless of the construction of homotopy groups. We employ two concrete examples to illustrate these results, highlighting the polarity reversal at fixed points.