# Linking invariant for the quench dynamics of a two-dimensional two-band Chern insulator

## TL;DR

This paper introduces a new topological invariant, the linking invariant in the -2c_i class, for the quench dynamics of a 2D two-band Chern insulator starting from a topological initial state, extending previous work on trivial initial states.

## Contribution

It proposes a modified Chern-Simons integral to define a linking invariant for nontrivial initial states in quench dynamics, which was not possible with traditional invariants.

## Key findings

- The linking invariant is given by a0(c_f - c_i) a0mod (2c_i).
- Concrete examples demonstrate the calculation of this invariant.
- The invariant captures the topological change during quench dynamics from a topological initial state.

## Abstract

We discuss the topological invariant in the (2+1)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number $c_i$), evolved by a post-quench Hamiltonian (with Chern number $c_f$). In contrast to the process with $c_i=0$ studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group $\pi_3(S^2)=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number of the initial spin configuration, which at the same time does not affect the (2+1)-dimensional topology. We show that the modified Chern-Simons integral gives rise to a topological invariant of this quench process, i.e., the linking invariant in the $\mathbb{Z}_{2c_i}$ class: $\nu = (c_f - c_i) \mod (2c_i)$. We give concrete examples to illustrate this result and also show the detailed deduction to get this linking invariant.