The quantum Hall effect encompasses a large variety of phases and phenomena arising from the combination of topological bands (the Landau levels), strong electron-electron interactions, and disorder. This thesis consists of an in-depth numerical study of some of these fascinating phenomena.
In the first part, we study the effect of geometric distortions of the underlying electron band on fractional quantum Hall (FQH) and composite Fermi liquid (CFL) states in the lowest Landau level (LLL). Through extensive density matrix renormalization group numerical simulations, we map the shape of correlations in both types of states in the presence of band mass anisotropy. We find that the geometry of FQH states depends on the LLL filling fraction, in agreement with a microscopic model of flux attachment. At half filling, the system forms a gapless CFL state with an emergent Fermi contour. We quantify its anisotropy and compare it to that of the zero-field carriers, finding an approximate square-root relationship between the two which is in excellent agreement with concurrent experiments on strained GaAs quantum wells. In contrast, we find the CFL Fermi contour is very weakly affected by other types of band distortions.
The second part of the thesis investigates quantum phase transitions in the LLL. We focus first on the integer quantum Hall plateau transition for disordered, non-interacting electrons. We study this transition in the presence of point impurities which remove a fraction of the states from the Landau band without altering its topological character. We then characterize the quasi-one dimensional limit of the transition, which reveals a surprising interplay between topology and disorder. We conclude with a study of quantum criticality in graphene Landau levels for clean, interacting electrons, focusing on a critical point between an antiferromagnet and a valence bond solid which is conjectured to exhibit `deconfined' quantum criticality.