  Topologically ordered systems are characterized by topological invariants
that are often calculated from the momentum space integration of a certain
function that represents the curvature of the many-body state. The curvature
function may be Berry curvature, Berry connection, or other quantities
depending on the system. Akin to stretching a messy string to reveal the number
of knots it contains, a scaling procedure is proposed for the curvature
function in inversion symmetric systems, from which the topological phase
transition can be identified from the flow of the driving energy parameters
that control the topology (hopping, chemical potential, etc.) under scaling. At
an infinitesimal operation, one obtains the renormalization group (RG)
equations for the driving energy parameters. A length scale defined from the
curvature function near the gap-closing momentum is suggested to characterize
the scale invariance at critical points and fixed points, and displays a
universal critical behavior in a variety of systems examined.
