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Hofstadter's Butterfly

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HofstadtersButterfly

Hofstadter's butterfly is the fractal set obtained by plotting the operator spectrum of the Harper equation

psi_(m+1)+psi_(m-1)+2cos(2pialpham+k)psi_m=Epsi_m,

against the magnetic flux parameter alpha, where k is a phase parameter. The difference equation models an electron on a two-dimensional periodic lattice in a perpendicular magnetic field. For rational alpha, the operator spectrum splits into finitely many bands. As alpha varies through rational and irrational values, the bands and gaps form the self-similar butterfly pattern. The spectrum has nontrivial scaling behavior (Thouless 1990), and its gaps carry topological quantum numbers called Chern numbers that determine the quantized Hall conductance (Avron and Osadchy 2001).

See also

Almost Mathieu Operator, Difference Equation, Fractal, Harper Equation, Operator Spectrum

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References

Avron, J. E. and Osadchy, D. "Butterflies and Topological Quantum Numbers." 20 Oct 2001. https://arxiv.org/abs/math-ph/0110026.Harper, P. G. "Single Band Motion of Conduction Electrons in a Uniform Magnetic Field." Proc. Phys. Soc. A 68, 874-878, 1955. https://doi.org/10.1088/0370-1298/68/10/304.Hofstadter, D. R. "Energy Levels and Wave Functions of Bloch Electrons in Rational and Irrational Magnetic Fields." Phys. Rev. B 14, 2239-2249, 1976. https://doi.org/10.1103/PhysRevB.14.2239.Thouless, D. J. "Scaling for the Discrete Mathieu Equation." Commun. Math. Phys. 127, 187-193, 1990. https://doi.org/10.1007/BF02096501.Trott, M. "Physical Remark: Hofstadter's Butterfly." In The Mathematica GuideBook for Numerics. New York: Springer-Verlag, pp. 321-329, 2006. https://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Hofstadter's Butterfly." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HofstadtersButterfly.html

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