  Motivated by the recent experimental breakthroughs in observing Fractional
Quantum Anomalous Hall (FQAH) states in moir\'e materials, we propose and study
various unconventional phase transitions between quantum Hall phases and Fermi
liquids or charge ordered phases upon tuning the bandwidth. At a fixed rational
lattice filling $\nu$, we describe a quantum Ginzburg-Landau theory to describe
the intertwinement between the FQAH and Charge Density Wave (CDW) orders. We
use this theory to describe phase transitions between the FQAH and a CDW
insulator. The critical theory for a direct second order transition resembles
that of the familiar deconfined quantum critical point (DQCP) but with an
additional Chern-Simons term. At filling -1/2, we study the possibility of a
continuous transition between the composite Fermi liquid (CFL) and the Fermi
liquid (FL) building on and refining previous work by Barkeshli and McGreevy.
Crucially we show that filling constraints ignored in that work ensure that
translation symmetry alone is enough to enable a second order CFL-FL
transition. We argue that there must be critical CDW fluctuations though
neither phase has long range CDW order. We present experimental signatures the
most striking of which is a universal jump of both longitudinal and Hall
resistivities at the critical point. With disorder, we argue that the CDW order
gets pinned and the CFL-FL evolution happens through an intermediate
electrically insulating phase with mobile neutral fermions. A clean analog of
this insulating phase with long range CDW order and a neutral fermi surface can
potentially also exist. We also present a critical theory for the CFL to FL
transition at filling -3/4. Our work opens up a new avenue to realize
deconfined criticality and fractionalized phases beyond familiar Landau level
physics in the moire Chern band system.
