Phase Transitions of Two-Dimensional Crystals in a Magnetic Field: Topological Perspectives on the Hofstadter Problem

Abstract

This thesis studies the electronic properties of 2d crystals in a perpendicular magnetic field using the tools of topological band theory. The energy spectrum of the system as it evolves adiabatically under increasing flux is known as the “Hofstadter Butterfly, ” a complex, fractal structure that is not captured by continuum descriptions or perturbative calculations. Motivated by this inadequacy, we develop a characterization of Hofstadter insulators via their zero field topology, which is readily calculable. We study a number of tight binding models, deriving their Hofstadter Hamiltonians and performing numerical experiments to understand what properties may protect a conducting phase transition, i.e. a valence band being lifted up to the Fermi energy at a critical value of the magnetic field. After outlining the relevant terminology and computational tools from topological band theory, as well as exemplifying them with analyses of some toy models, we present a simple proof of gap closing in Chern insulators using the magnetic translation group. From the numerics, we see that a nonzero Chern number is not always necessary, and we prove a stronger characterization of magnetic phase transitions using inversion symmetry. This result anticipates extensions. In the final section, we study the fragile topology of bilayer graphene and demonstrate which symmetries protect a gapless phase in magnetic field.