you should look for " berry phase".

wikipedia gives quite good answers in this case.

i can also recommend the book "Modern Quantum Mechanics" by J.J. Sakurai for a more detailed explanation.

If you still want a specific example after reading the suggested materials given by Merle, you can take a look at Na3. This molecule has 3 vibrational degrees of freedom, one A mode that is the symmetric stretch of all 3 Na bonds, and an E mode of two degenerate anti-symmetric stretches. If you name the two E normal mode vectors x and y, then R=R0+δ*(x*sin[t]+y*cos[t]) give you a path that rotates around the intersection, when you go from t=0 to t=2π. Here δ is any small number that gives the radius of the path. This is the same for a general conical intersection, where x and y are the "branching space" parameters. These vectors define the two directions along which the degeneracy is lifted linearly. Potential energy surfaces look like double cones in the 2D branching space, hence the name conical intersection. You can set up a model Hamiltonian and follow through the eigenvectors to see the sign change. Of course, instead of drawing a perfect circle, you can draw any arbitrary path that surrounds the intersection exactly once, and the phase remains the same. Effect of such phase can be observed in vibrational spectrum of the species.

Thanks for your responses. To be more clear, I want to know how this rotations happens in molecular dynamics. For example, talking about Xiaolei's example, how is this path (named R) formed? Actually, I'm talking about the reality of this phenomenon. In nature or lab, does sign change happen due to interaction with other molecules or environment? What about obervables? and also is there any experimental procedure to see this sign change?

I suppose by molecular dynamics you mean the dynamic behavior of a molecular system, instead of the computational method that bear that name. If so, then yes, geometric phases have many dynamical consequences. Geometric phase is a topological property of the electronic wave function and is a result of (the breakdown of) Born-Oppenheimer approximation. Since total wave function of nuclear and electron is single valued, geometric phase dictates the symmetry of the nuclear wave function and as a result changes dynamical behaviors. Its like the Pauli principle, where exchange of electrons change sign of a wave function, which serves as a constraint on what kind of wave functions are valid. Alternatively, dynamics calculations can be performed in a diabatic representation where the PES are smooth and geometric phases vanish. If you don't do either of these two things, you will never be able to calculate the correct vibrational spectrum of Na3. Just think about a Bohr model hydrogen atom. If you require that the electron change sign when it goes around the nuclear for a full circle, instead of having the same value, then the energy levels will be completely different.