Bulk spectrum and K-theory for infinite-area topological quasicrystals

Abstract

The bulk spectrum of a possible Chern insulator on a quasicrystalline lattice is examined. The effect of being a 2D insulator seems to override any fractal properties in the spectrum. We compute that the spectrum is either two continuous bands, or that any gaps other than the main gap are small. After making estimates on the spectrum, we deduce a finite system size, above which the K-theory must coincide with the K-theory of the infinite system. Knowledge of the spectrum and K-theory of the infinite-area system will control the spectrum and K-theory of sufficiently large finite systems. The relation between finite volume K-theory and infinite volume Chern numbers is only proven to begin, for the model under investigation here, for systems on Hilbert space of dimension around 17 × 106. The real-space method based on the Clifford spectrum allows for computing Chern numbers for systems on Hilbert space of dimension around 2.7 × 106. New techniques in numerical K-theory are used to equate the K-theory of systems of different sizes.