The Hofstadter Butterfly and some physical consequences.

Opening its beautiful wings for the first time four decades ago, the Hofstadter Butterfly emerged as a self-similar pattern of bands and gaps displaying the allowed energies for two dimensional crystalline electrons in a perpendicular magnetic field.$^{\mathrm{1\thinspace }}$Within the Harper model, as the external field parameter is varied well defined gaps traverse the spectrum, some closing at a Dirac point where two approaching bands touch. Such band edges degeneracy is lifted in more realistic models.$^{\mathrm{2}}$ Gaps have a unique label that determines the Hall conductivity of a noninteracting electron system, as observed in recent experiments.$^{\mathrm{3}}$ When the 2D electron assembly is allowed to interact in the absence of an underlying periodic potential, the mean field approximation predicts a liquid at integer filling fractions and electron density fluctuations otherwise, which if periodic may be represented again by a Harper equation. The intriguing odd denominator rule observed in experiment in the fractional quantum Hall regime is then a natural prediction of the model.$^{\mathrm{4}}$ \begin{enumerate} \item D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976) \item F. Claro, Phys. Status Solidi (b) 104, K31 (1981) \item C. R. Dean et al, Nature 497, 598 (2013) \item F. Claro, Phys. Rev. B 35, 7980 (1987) \end{enumerate}