Phase transitions out of quantum Hall states in moiré materials

Motivated by the recent experimental breakthroughs in observing fractional quantum anomalous Hall (FQAH) states in moiré 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 $\nu =-1/2$, 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 [Phys. Rev. B 86, 075136 (2012)]. 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 discuss the properties of this phase and the nature of its phase transitions. We also present a critical theory for the CFL to FL transition at filling $\nu =-3/4$. Our work opens up a new avenue to realize deconfined criticality and fractionalized phases beyond familiar Landau level physics in the moiré Chern band system.