Quench dynamics for a two-dimensional two-band model starting with a topological initial state

We discuss the dynamical process of a two-dimensional two-band system starting with a topologically nontrivial initial state, 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 classified by the Hopf invariant that is described by the homotopy group π₃(S²)=Z. It is possible, however, to calculate the Chern-Simons integral with a complementary part to cancel the Chern number of initial spin configuration. We show the Chern-Simons integral with the complementary part is the topological invariant of this process, which is a linking invariant in the Z_2ci class: ν = (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.