Fractional charge and inter-Landau level states at points of singular curvature

Abstract: The quest for universal signatures of topological phases is fundamentally important since these properties are robust to variations in system-specific details. Here we present general results for the response of quantum Hall states to points of singular curvature in real space. Such topological singularities may be realized, for instance, at the vertices of a cube, the apex of a cone, etc. We find, using continuum analytical methods, that the point of curvature binds an excess fractional charge. In addition, sequences of states split away, energetically, from the degenerate bulk Landau levels. Importantly, these inter-Landau level states are bound to the topological singularity and have energies that are $\emph{universal}$ functions of bulk parameters and the curvature. Remarkably, our exact diagonalization of lattice tight-binding models on closed manifolds shows that these results continue to hold even when lattice effects are significant, where the applicability of continuum techniques could not have been justified a priori. Moreover, we propose how these states may be readily experimentally actualized. An immediate technological implication of these results is that these inter-Landau level states, being as they are $\emph{both}$ energetically and spatially isolated quantum states, are promising candidates for constructing qubits for quantum computation.