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The Hofstadter Butterfly

Spectrum The Hofstadter butterfly (on the left) is the energy spectrum of an electron, restricted to move in two-dimensional periodic potential under the influence of a perpendicular magnetic field. The horizontal axis is the energy and the vertical axis is the magnetic flux through the unit cell of the periodic potential. The flux is a dimensionless number when measured in quantum flux units (will call it a). It is an example of a fractal energy spectrum. When the flux parameter a is rational and equal to p/q with p and q relatively prime, the spectrum consists of q non-overlapping energy bands, and therefore q+1 energy gaps (gaps number 0 and q are the regions below and above the spectrum accordingly). When a is irrational, the spectrum is a cantor set.
The gaps in the spectrum correspond to integer values of the Hall conductance. The figure on the right shows the gaps, color coded according to the Hall conductance. The warm colors represent positive values of Hall conductance, and the cold colors represent negative values. Zero Hall conductance is left blank. Colored spectrum

Some facts about the Hofstadter Butterfly:

For rational values of a = p/q all the gaps are open, except the middle gap for even q.
For a = p/q, the Hall conductance s corresponding to the gap number r satisfies the diophantine equation p s + q m = r. The unique solution is chosen such that |s| < q / 2.
The number of disjoint regions in the colored figure having the (same) color corresponding to the Hall conductance s is given by:
where the function is the Euler function which counts the number of positive integers not greater than j and relatively prime to j.
This result assumes that for irrational values of a all the gaps are open (see open questions below).
The edges of gap number r at rational value of a = p/q, are common points for infinitely many regions. The set of Hall conductances s of these regions is exactly the set of solutions to the diophantine equation p s + q m = r.

Still open questions:

Are all the gaps open for irrational values of a ?
For almost all values of a (having full Lebesgue measure), the spectrum is a cantor set of zero Lebesgue measure. Is this true for all irrational values of a ?