2 edited tags
| link
    Tweeted twitter.com/StackMma/status/960291380466782208
1
source | link

Tracking Eigenvalues Through a Crossing

Suppose I have a matrix which depends on some parameter. I want to compute the eigenvalues as a function of this parameter, and then plot them. For example, I may have a matrix representing the Hamiltonian of a system in a magnetic field, and I want to plot the energies of the various states as a function of B-field. Sometimes the eigenvalues may cross each other, but I want to make sure the right eigenvalue stays associated with its own state. For example, consider the matrix

  H = {{1/2 + 21 B, 0, 0, 0, 0, 0}, {0, 1/2 + 7 B, 0, 0, 7 Sqrt[2] B, 
      0}, {0, 0, 1/2 - 7 B, 0, 0, 7 Sqrt[2] B}, {0, 0, 0, 1/2 - 21 B, 0, 
      0}, {0, 7 Sqrt[2] B, 0, 0, -1 + 14 B, 0}, {0, 0, 7 Sqrt[2] B, 0, 
      0, -1 - 14 B}}

If I evaluate evals = Eigenvalues[H] and then plot the result, I see a nice plot of the eigenvalues and the states follow properly through crossings, for example just below B = 0.05. Good crossing

As my matrix gets large, it becomes extremely slow to do the eigendecomposition analytically, so I'd rather do it numerically. However, in this case the eigenvalues don't properly track through crossings-- instead, they are sorted by value from largest to smallest in absolute value. For example, if I do the following

list = {};
Do[
 evalsN = Eigenvalues[H];
 AppendTo[list, evalsN],
 {B, 0, 0.1, 0.001}]

then I get a messed up plot like this:

Bad plot

This is for two reason: (1) because the ordering is all off, and (2) because the states don't track through crossing. I can fix problem (1) by ordering the eigenvalues by their magnitude, but that does NOT help them track through a crossing (e.g., see how the red and purple lines don't cross through one another around element 50). How do I do the second task?