  We study a special class of topological phase transitions in two dimensions
described by the inversion of bands with relative angular momentum higher than
1. A band inversion of this kind, which is protected by rotation symmetry,
separates the trivial insulator from a Chern insulating phase with higher Chern
number, and thus generalizes the quantum Hall transition described by a Dirac
fermion. Higher angular momentum band inversions are of special interest, as
the non-vanishing density of states at the transition can give rise to
interesting many-body effects. Here we introduce a series of minimal lattice
models which realize higher angular momentum band inversions. We then consider
the effect of interactions, focusing on the possibility of electron-hole
exciton condensation, which breaks rotational symmetry. An analysis of the
excitonic insulator mean field theory further reveals that the ground state of
the Chern insulating phase with higher Chern number has the structure of a
multicomponent integer quantum Hall state. We conclude by generalizing the
notion of higher angular momentum band inversions to the class time-reversal
invariant systems, following the scheme of Bernevig-Hughes-Zhang (BHZ). Such
band inversions can be viewed as transitions to a topological insulator
protected by rotation and inversion symmetry, and provide a promising venue for
realizing correlated topological phases such as fractional topological
insulators.
