Type-II pumping beyond resonance principle: From energetic to geometric rules

Conventionally, pumping relies on energetic resonance: energy quanta ${\hbar}{\omega}$ matches the gap $\Delta$. Under linear approximation, this is known as the Fermi golden rule (FGR). However, this principle becomes challenging to apply in the "0/0" limit, where ${\omega},{\Delta}{\rightarrow}0$ simultaneously. In "0/0" scenarios, such as topological phase transition (TPT), a type-II pumping, geometric pumping (GP), is recognized subject to geometric rules, distinguished from type-I dictated by FGR. Type-I features an "arrow of energy", sending particles higher in energy, reflected by FGR's dependence on Fermi distribution $f_v-f_c$ (probabilities of valence and conduction bands). While GP is non-directional, its probability relies on $f_v+f_c-2f_v f_c$ instead, a key signature for detection. In this work, we address: (1) the concept of GP; (2) its features of fractionality, irreversibility, and dependence on TPT; (3) experimental detection with ultra-fast spectrum in coherent phonon driving of ZrTe$_5$.