Geometry, stability and response in lattice quantum Hall systems
Integer and fractional quantum Hall phases are prototypical examples of topologically ordered phases of matter. A standard setting for studying these phases is in continuum two-dimensional electron fluids with broken time-reversal symmetry, where the single particle bands are familiar Landau levels. In this thesis, we explore aspects of quantum Hall fluids in the presence of periodic lattice potential.
We focus mostly on the perturbative, Harper-Hofstadter regime of low magnetic flux density, where the single-particle bands are effectively mixtures of Landau levels. We study the stability of strongly-correlated fractional quantum Hall fluids in relation to the geometry of single-particle Chern bands in tight-binding models. We first treat the case of the Hofstadter model and show the convergence of the geometry of Hofstadter bands to that of Landau levels in the limit of small flux density. We then introduce and study a new series of tight-binding models which do not necessarily converge to Landau levels but still host fractional phases. In each case, we observe correlations between band geometry and many-body gaps consistent with the “geometric stability hypothesis.”
Finally, we give a perturbative calculation of the current response of Chern bands to inhomogeneous electric fields. In continuum quantum Hall fluids with galilean symmetry, the O(q2) contribution to this response is related to the Hall viscosity, which measures the linear response to variations in spatial geometry in time-reversal odd two-dimensional fluids.