# Hofstadter's butterfly

In physics, **Hofstadter's butterfly** describes the spectral properties of non-interacting two dimensional electrons in a magnetic field. The fractal, self-similar, nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter^{[1]} and is one of the early examples of computer graphics. The name reflects the visual resemblance of the figure on the right to swarm of butterflies flying to infinity, it is one of the rare non-random fractal structures in physics, along with KAM tori.^{[citation needed]} The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect, and D.J. Thouless has been awarded the Nobel prize in physics in 2016 for the discovery^{[2]} that the wings of the butterfly are characterized by Chern integers^{[3]}, the quantized Hall conductances discovered in 1980 by Klaus von Klitzing for which he has been awarded the Nobel prize in 1985. The colors in the diagram reflect the different Chern numbers.

## Background[edit]

Hofstadter described the structure in 1976 in an article on the energy levels of Bloch electrons in magnetic fields,^{[1]} it gives a graphical representation of the spectrum of the almost Mathieu operator for at different frequencies. The intricate mathematical structure of this spectrum was discovered by Soviet physicist Mark Ya. Azbel' in 1964.^{[4]} However, Azbel' did not plot the structure as a geometrical object.

Written while Hofstadter was at the University of Oregon, his paper was influential in directing further research. Hofstadter predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional square lattice, as a function of a magnetic field applied to the system, formed what is now known as a fractal set; that is, the distribution of energy levels for small scale changes in the applied magnetic field recursively repeat patterns seen in the large-scale structure.^{[1]} "Gplot", as Hofstadter called the figure, was described as a recursive structure in his 1976 article in *Physical Review B*,^{[1]} written before Benoit Mandelbrot's newly coined word "fractal" was introduced in an English text. Hofstadter also discusses the figure in his 1979 book *Gödel, Escher, Bach*; the structure became generally known as "Hofstadter's butterfly".

## Confirmation[edit]

In 1997 the Hofstadter butterfly was reproduced in experiments with microwave guide equipped by an array of scatterers.^{[5]} Similarity of the mathematical description of the microwave guide with scatterers and of Bloch's waves in magnetic field allowed to reproduce the Hofstadter butterfly for periodic sequences of the scatterers.

In 2013, three separate groups of researchers independently reported evidence of the Hofstadter butterfly spectrum in graphene devices fabricated on hexagonal boron nitride substrates.^{[6]}^{[7]}^{[8]} In this instance the butterfly spectrum results from interplay between the applied magnetic field, and the large scale moiré pattern that develops when the graphene lattice is oriented with near zero-angle mismatch to the boron nitride.

In September 2017, John Martinis group in Google, in collaboration with Angelakis group in CQT Singapore, published results from a simulation of 2D electrons in a magnetic field using interacting photons in 9 superconducting qubits; the simulation recovered Hofstadter's butterfly, as expected.^{[9]}

## References[edit]

- ^
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