  The correlation functions related to topological phase transitions in
inversion-symmetric lattice models described by $2\times 2$ Dirac Hamiltonians
are discussed. In one dimension, the correlation function measures the
charge-polarization correlation between Wannier states at different positions,
while in two dimensions it measures the itinerant-circulation correlation
between Wannier states. The correlation function is nonzero in both the
topologically trivial and nontrivial states, and allows to extract a
correlation length that diverges at topological phase transitions. The
correlation length and the curvature function that defines the topological
invariants are shown to have universal critical exponents, allowing the notion
of universality classes to be introduced. Particularly in two dimensions, the
universality class is determined by the orbital symmetry of the Dirac model.
The scaling laws that constrain the critical exponents are revealed, and are
predicted to be satisfied even in interacting systems, as demonstrated in an
interacting topological Kondo Insulator.
