Harvard University,FAS
Fall 2003

Mathematics Math21b
Fall 2003

Linear Algebra and Differential Equations

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

The Hofstadter Butterfly

(An example, where determimants appear in physics)
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.
See the Java applet.


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