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The Hofstadter Butterfly PDF Print E-mail
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Applet - General
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Thursday, 29 May 2008 00:00

The picture below illustrates a function f(x,y) of two variables which is defined using determinant of matrices. For rational y=p/q form f(x,y) = log[ | det (L(y) -x) | ]/q, where L(y) is a (q x q) matrix with side diagonal entries 1 and diagonal entries V(k) = 2 cos(2 pi k p/q ):

      |   V(1)     1      0    ...    0        1     |
      |     1    V(2)     1    ...   ...       0     |
L(y)= |     0      1     ...   ...   ...      ...    |
      |    ...    ...    ...   ...    1        0     |
      |     0     ...    ...    1    V(q-1)    1     |
      |     1      0     ...    0     1       V(q)   |
Physically, the x-coordinate is the energy. The y coordinate is related to a magnetic flux.
The Hofstadter butterfly is the set, where f(x,y)=0. The picture to the right is colored according to the value of f(x,y). Mathematicians call the function f(x,y) a Lyapunov exponent. It is also defined for irrational y through a limit.

 



The Java applet is adapted from Wolfgang Kinzel/Georg Reents,"Physics by Computer" Springer Press (1998)

                            http://www.math.harvard.edu/archive/21b_fall_03/hofstadter/index.html
 


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Last Updated on Friday, 30 May 2008 12:10