A generic approach to extract geometry from non-adiabatic (gap-closing) processes and its consequences: geometric pumping and dephasing at topological phase transitions in topological insulators or semi-metals.

Geometry in physics arises from adiabaticity via an analog to “parallel transport”. Locally, geometry is reflected by Berry curvatures, from which one can deduce various observables (e.g., adiabatic pump, electric polarization). 

Our first result (Song 2021) is to show Berry’s framework (and its development) is not the unique one to incorporate geometry. In fact, parallel transport or local curvatures are dispensable. Our scheme relies on making quantum evolution mimic classical trajectories via a measure-preserving formalism (MPF). Measure preservation means a conservative system’s volume in phase space {p,q} is constant during evolution, known as the Liouville theorem. However, its quantum counterpart, Wigner flow, is compressible, which depreciates usage and poses a long-pending question: if MPF can be rigorously constructed and significantly used in quantum scenarios.

The second result is about an observable, inter-band pumping, namely geometric pumping. Its rate purely depends on an angle parameter, which, in certain in- stances, is linked to Z₂. Since the pumping is for bulk, it challenges the wisdom that Z₂ only implies sur- face/interface, known as bulk-edge correspondence. It relies on the flipping of Z₂ (not which state of Z₂) thus is a genuine consequence of topological phase transitions (TPT). We set out from a spin model to introduce a generic MPF, with which pumping rates and conditions are proved. Meanwhile, rational/irrational numbers are giving subtle influence. This work helps account for recent experiments on TPT in ZrTe₅ (Luo 2021, Vaswani 2020).