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MENTAL MODEL · M4460

Hofstadter's Butterfly

Hofstadter's Butterfly
TechnicalHigh supportQuantum Mechanics
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Version 1.0.0 · Updated 2026-07-28

CORE DEFINITION

In condensed matter physics, Hofstadter's butterfly refers to the butterfly-like pattern exhibited by the energy spectrum of electrons in a two-dimensional lattice under the combined influence of a periodic potential and an external magnetic field. In 1976, Hofstadter described the fractal and self-similar nature of this energy spectrum in his doctoral thesis; this spectrum diagram has also become one of the early examples of modern scientific data visualization. As Hofstadter wrote: "The large gaps in the figure form a very striking pattern, somewhat resembling a butterfly," hence the name "Hofstadter's butterfly." Hofstadter's butterfly plays a pivotal role in the theory of the integer quantum Hall effect and topological quantum numbers.

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In condensed matter physics, Hofstadter's butterfly refers to the butterfly-like pattern exhibited by the energy spectrum of electrons in a two-dimensional lattice under the combined influence of a periodic potential and an external magnetic field. In 1976, Hofstadter described the fractal and self-similar nature of this energy spectrum in his doctoral thesis; this spectrum diagram has also become one of the early examples of modern scientific data visualization.

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Hofstadter's butterfly is a self-similar fractal diagram of the band structure of two-dimensional electrons in a magnetic field, revealing the complex energy levels in the quantum Hall effect.

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    zh.wikipedia.orghttps://zh.wikipedia.org/wiki/%E9%9C%8D%E5%A4%AB%E6%96%BD%E5%A1%94%E7%89%B9%E8%9D%B4%E8%9D%B6ZH · Explicit
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