  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.
