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Let's say I have a 2-state system described by a $2\times 2$ non-degenerate Hamiltonian in some 2D parameter space. This is in the context of condensed matter, but should be more fundamental quantum physics. Then, for numerical investigation, I can choose a closed loop in 2D parameter space and track the evolution of the 2 instantaneous eigenstates $|u_1\rangle,|u_2\rangle$ of the model. Then, when I visualize the eigenstates along the loop for instance by plotting $L=(\text{Re}|u_1\rangle,\text{Im}|u_1\rangle)$, I sometimes get a self-intersecting $L$. What are the consequences of this, and why does this happen? I usually see only the amplitude squared of wavefunctions being used (for probability), but for some reason, this kind of 'degeneracy' feels wrong (especially because the system is non-degenerate a priori). I haven't been able to find anything about this in the literature yet.

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