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| [[Image:Hofstadter's butterfly.png|thumb|Hofstadter's butterfly]]
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| In [[physics]], '''Hofstadter's butterfly''' is a mathematical object describing the theorised behaviour of [[electron]]s in a [[magnetic field]]. It was discovered by [[Douglas Hofstadter]] in 1976, who later abandoned physics research and became notable as a [[computer scientist]] and author. It takes its name from its visual resemblance to a [[butterfly]]. It is a [[fractal]] structure and as such it shows [[self-similarity]], meaning that small fragments of the structure contain a (distorted) copy of the entire structure. It is one of the rare fractal structures discovered in [[physics]], along with [[Kolmogorov–Arnold–Moser theorem|KAM tori]].
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| To test whether Hofstadter's butterfly describes real electron behaviour requires accurate measurements. Such were not possible when he wrote his paper. However, more recent experimental research has confirmed the characteristic butterfly shape. | |
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| == Background ==
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| Hofstader described the structure in 1976 in an article on the [[energy level]]s of [[Bloch electron]]s in magnetic fields.<ref name="energylevels">{{cite journal|author=[[Douglas R. Hofstadter]] |year=1976 |title= Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields |journal=[[Physical Review B]] |volume=14 |issue=6 |pages= 2239–2249 |doi= 10.1103/PhysRevB.14.2239|bibcode = 1976PhRvB..14.2239H }}</ref> It gives a graphical representation of the spectrum of the [[almost Mathieu operator]] for <math>\lambda = 1</math> at different frequencies,
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| 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 [[pattern]]s seen in the large-scale structure.<ref name="energylevels"/> "Gplot", as Hofstadter called the figure, was described as a [[recursive structure]] in his 1976 article in ''[[Physical Review B]]'',<ref name="energylevels"/> 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".
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| == Confirmation ==
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| [[Image:Gplot by Hofstadter.jpg|thumb|Rendering of the butterfly by Hofstadter]]
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| It has recently been experimentally confirmed in transport measurements in [[2DEG|two-dimensional electron systems]] with a superimposed nano-fabricated lattice.<ref>[http://physics.technion.ac.il/~odim/hofstadter.html The Hofstadter Butterfly]</ref>{{Failed verification|date=April 2010}} In 2013, it was shown to exist in an association of graphene and boron nitride.<ref>[http://phys.org/news/2013-05-graphene-butterflies.html Catching graphene butterflies], phys.org, May 2013</ref> In May 2013, two separate but cross-collaborating research teams reported the first ever observations of the Hofstadter butterfly: a team from the [[High Magnetic Field Laboratory]] (MagLab) at [[Florida State University]]; and one that included researchers from [[Columbia University]], [[City University of New York]], the [[University of Central Florida]], [[Tohoku University]] and the [[National Institute for Materials Science]] in Japan.<ref>[http://planetsave.com/2013/05/15/long-predicted-fractal-energy-pattern-observed-for-first-time-by-physicists/ Long predicted fractal energy pattern observed for first time by physicists], Planetsave.com, May 15, 2013</ref>
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| ==References==
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| {{reflist}}
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| {{Douglas Hofstadter}}
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| [[Category:Fractals]]
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| [[Category:Condensed matter physics]]
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