A Crossing degeneracies and geometrical phases

[andresB]

Assume all the usual things for the usual things for the geometric phases: A Hamiltonian that depend on external parameters, Adiabatic evolution, cyclic evolution in parameter space and all that

If through the evolution in parameter space there is no energy level crossing, then a eigenvector of the hamiltonian will acquire a geometric phase; the Berry phase for non-degenerate levels and the Wilzeck-Zee phase otherwise.

Now, what happen when there is a crossing in a energy level?. Suppose we start the adiabatic evolution in point of parameter space that have an accidental degeneracy and that after we leave that point the degeneracy is completely lifted. What happen after a cyclic evolution?

 

[azntoon]

The answer is that the geometric phase acquired depends on the exact path of parameters taken. In particular, if the paths are such that at every point of the cycle the energies of those two levels are exchange, then the geometric phase will be zero. This is because the eigenvectors of the Hamiltonian at the end of the cycle will be the same as the eigenvectors at the beginning in this case, and no geometric phase can be accumulated.