Measuring Topological Number of a Chern-Insulator from Quench Dynamics

TL;DR

This paper demonstrates a method to measure the topological invariant of a Chern insulator through dynamical quench processes, linking the linking number of phase vortex trajectories to the Chern number.

Contribution

It introduces a novel approach to determine the topological number from quench dynamics using the Hopf invariant and linking number in a two-band Chern insulator.

Findings

Linking number of phase vortex trajectories equals the Chern number.

Method applied to experimental data from cold atom systems.

Phase boundary of the Hamiltonian can be identified via linking number.

Abstract

In this letter we show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two-dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the $[k_x,k_y,t]$ space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the $[k_x,k_y,t]$ space, and the linking number of these two trajectories exactly equals to the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.