The Hofstadter Butterfly



For rational y=p/q, define
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 the magnetic flux.
The Hofstadter butterfly is the set, where f(x,y)=0. The picture 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.
Click on the picture to see a high resolution version with 1900x1900 pixels. It has been computed in Mathematica by going up to q=1223.
Hofstadter Butterlfly


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