Hofstadter's butterfly

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Hofstadter's Butterfly refers to a fractal discovered by Douglas Hofstadter in his paper "Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields." Hofstadter's Butterfly was the first fractal to be formulated in physics. It gives a graphical version of the spectrum of the almost Mathieu operator for $λ = 1$ at different frequency.

Written while he was at the University of Oregon, this paper was influential in directing further research. Hofstadter predicted that the allowed energy level values of an electron in this crystal lattice, as a function of a magnetic field applied to the system, formed a fractal set. That is, the distribution of energy levels for large scale changes in the applied magnetic field repeat patterns seen in the small scale structure. This fractal structure is generally known as "Hofstadter's butterfly", and has recently been confirmed in transport measurements in two-dimensional electron systems with a superimposed nano-fabricated lattice.^[1]^[2]