The Hofstadter model and other fractional chern insulators

Fractional Chern insulators (FCIs) are strongly correlated, topological phases of matter that may exist on a lattice in the presence of broken time-reversal symmetry. This thesis explores the link between FCI states and the quantum Hall effect of the continuum in the context of the Hofstadter model, using a combination of nonperturbative, perturbative and numerical methods. We draw links to experimental realisations of topological phases, and go on to consider a novel way of generating general FCI states using strong interactions on a lattice. We begin by considering the Hofstadter model at weak field, where we use a semiclassical analysis to obtain nonperturbative expressions for the band structure and Berry curvature of the single-particle eigenstates. We use this calculation to justify a perturbative approximation, an approach that we extend to the case when the amount of flux per plaquette is close to a rational fraction with a small denominator. We find that eigenstates of the system are single- or multicomponent wavefunctions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalised Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions, and magnetic field strengths. We present numerical simulations of the Hofstadter model relevant to its recent experimental realisation using optical lattices, noting the additional complications that arise in the presence of an external trap. Finally, we show that even if a noninteracting system is topologically trivial, it is possible to stabilise an FCI state by introducing strong interactions that break time-reversal symmetry. We show that this method may also be used to create a (time-reversal symmetric) fractional topological insulator, and provide numerical evidence to support our argument.