Topological phase transitions between bosonic and fermionic quantum Hall states near even-denominator filling factors

We study the quantum critical point between the fermionic $\nu=8$ quantum Hall state and the bosonic $\nu=2$ quantum Hall state of Cooper pairs. Our study is motivated by the composite fermion construction for the daughter states of even-denominator fractional quantum Hall states and the experimentally observed transition between the daughter and the Jain states at the same filling. We show that this transition is equivalent to the transition between a neutral invertible $E_8$ state and a topologically trivial state. These transitions can be described in a partonic framework as a cascade of mass changes of four neutral Dirac fermions coupled to multiple Abelian Chern--Simons $U(1)$ gauge fields. In the absence of fine-tuning, the transition is split into a series of at least eight distinct transitions, with at least seven distinct intermediate topologically ordered phases that host neutral anyons.