Killing the Hofstadter butterfly, one bond at a time

Abstract

Electronic bands in a square lattice when subjected to a perpendicular magnetic field form the Hofstadter butterfly pattern. We study the evolution of this pattern as a function of bond percolation disorder (removal or dilution of lattice bonds). With increasing concentration of the bonds removed, the butterfly pattern gets smoothly decimated. However, in this process of decimation, bands develop interesting characteristics and features. For example, in the high disorder limit, some butterfly-like pattern still persists even as most of the states are localized. We also analyze, in the low disorder limit, the effect of percolation on wavefunctions (using inverse participation ratios) and on band gaps in the spectrum. We explain and provide the reasons behind many of the key features in our results by analyzing small clusters and finite size rings. Furthermore, we study the effect of bond dilution on transverse conductivity($\sigma_{xy}$). We show that starting from the clean limit, increasing disorder reduces $\sigma_{xy}$ to zero, even though the strength of percolation is smaller than the classical percolation threshold. This shows that the system undergoes a direct transition from a integer quantum Hall state to a localized Anderson insulator beyond a critical value of bond dilution. We further find that the energy bands close to the band edge are more stable to disorder than at the band center. To arrive at these results we use the coupling matrix approach to calculate Chern numbers for disordered systems. We point out the relevance of these results to signatures in magneto-oscillations.