Unification and measurement of topological order

Topological order manifests as different physical quantities according to the dimensions and symmetries of the materials, such as Majorana fermions and quantized Hall conductance, just to list a few. We elaborate that all these phenomena in any dimension and symmetry class can be described by a unified topological invariant called wrapping number, which is analogous to the Gauss map in differential geometry. The wrapping number takes the form of integration of a certain curvature function over the momentum space, and the curvature function is further shown to be equivalent to an important quantity that generally characterizes all quantum phase transitions called fidelity susceptibility, also known as the quantum metric. A loss-fluence spectroscopy is further proposed to measure the quantum metric in a pump-probe experiment, through which the topological order in any noninteracting system can be directly measured, as demonstrated explicitly for graphene.