I have been mainly exposed to 2D momentum-space condensed matter models in the context of Berry-related topology. I now want to study models where, if I take a closed loop in momentum space, I will have the possibility of an interband transition governed by selection rules. So far, all I have seen are charge-pumping-like effects with the Rice-Mele model (quantized flux pumping after one closed loop trajectory; example). I have seen some 2D models for excitons as well (example), but these don't consider a loop explicitly. Additionally, I understand that there is a perspective of potential virtual transitions associated with Berry curvature (due to its inherent interband character). However, I am looking for something more concrete than that.
I have read a bit about Landau-Zener transitions, but most works I have seen involve the time variable explicitly (example of selection rules). However, I am looking for an example of a model that studies interband transitions and selections rules in the context of a closed loop in k-space (very likely parameterized by parameter $t$: $(k_x(t),k_y(t))$. Since I have a ways to go in familiarizing myself with the field, I was hoping someone could suggest a resource that features such a model.
