  We develop a method to characterize topological phase transitions for
strongly correlated Hamiltonians defined on two-dimensional lattices based on
the many-body Berry curvature. Our goal is to identify a class of quantum
critical points between topologically nontrivial phases with fractionally
quantized Hall (FQH) conductivity and topologically trivial gapped phases
through the discontinuities of the many-body Berry curvature in the so-called
flux Brillouin zone (fBZ), the latter being defined by imposing all possible
twisted boundary conditions. For this purpose, we study the finite-size
signatures of several quantum phase transitions between fractional Chern
insulators and charge-ordered phases for two-dimensional lattices by evaluating
the many-body Berry curvature numerically using exact diagonalization. We
observe degeneracy points (nodes) of many-body energy levels at high-symmetry
points in the fBZ, accompanied by diverging Berry curvature. We find a
correspondence between the number and order of these nodal points, and the
change of the topological invariants of the many-body ground states across the
transition, in close analogy with Weyl nodes in non-interacting band
structures. This motivates us to apply a scaling procedure, originally
developed for non-interacting systems, for the Berry curvature at the nodal
points. This procedure offers a useful tool for the classification of
topological phase transitions in interacting systems harboring FQH-like
topological order.
