Measurement of Topological Order based on Metric-Curvature Correspondence

A unified expression for topological invariants has been proposed recently to describe the topological order in Dirac models belonging to any dimension and symmetry class. We uncover a correspondence between the curvature function that integrates to this unified topological invariant and the quantum metric that measures the distance between properly defined many-body Bloch states in momentum space. Based on this metric-curvature correspondence, a time-resolved and angle-resolved photoemission spectroscopy experiment is proposed to measure the violation of spectral sum rule caused by a pulse electric field to detect the quantum metric, from which the topological properties of the system may be extracted.