Dynamical topological invariant after a quantum quench

TL;DR

This paper introduces a method to define a dynamical topological invariant for one-dimensional topological systems after a quantum quench, linking it to the initial and final Hamiltonian topologies.

Contribution

It presents a novel way to characterize post-quench dynamics using a topological invariant on a momentum-time manifold, specifically for two-band topological insulators.

Findings

Dynamical topological invariant can be defined on each $S^2$ submanifold.

The invariant relates to the difference between initial and final Hamiltonian topologies.

Explicit calculations demonstrate the geometrical interpretation of the invariant.

Abstract

We show how to define a dynamical topological invariant for general one-dimensional topological systems after a quantum quench. Focusing on two-band topological insulators, we demonstrate that the reduced momentum-time manifold can be viewed as a series of submanifold $S^2$, and thus we are able to define a dynamical topological invariant on each of the sphere. We also unveil the intrinsic relation between the dynamical topological invariant and the difference of topological invariant of the initial and final static Hamiltonian. By considering some concrete examples, we illustrate the calculation of the dynamical topological invariant and its geometrical meaning explicitly.