# When/why does an evolving wavefunction loop intersect with itself?

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.