  For D-dimensional weakly interacting topological insulators in certain
symmetry classes, the topological invariant can be calculated from a D- or
(D+1)-dimensional integration over a certain curvature function that is
expressed in terms of single-particle Green's functions. Based on the
divergence of curvature function at the topological phase transition, we
demonstrate how a renormalization group approach circumvents these integrations
and reduces the necessary calculation to that for the Green's function alone,
rendering a numerically efficient tool to identify topological phase
transitions in a large parameter space. The method further unveils a number of
statistical aspects related to the quantum criticality in weakly interacting
topological insulators, including correlation function, critical exponents, and
scaling laws, that can be used to characterize the topological phase
transitions driven by either interacting or noninteracting parameters. We use
1D class BDI and 2D class A Dirac models with electron-electron and
electron-phonon interactions to demonstrate these principles, and find that
interactions may change the critical exponents of the topological insulators.
