Geometric phase and fractional orbital-angular-momentum states in electron vortex beams

We study here fractional orbital-angular-momentum (OAM) states in electron vortex beams (EVBs) from the perspective of the geometric phase. We consider the skyrmionic model of an electron, where it is depicted as a scalar electron orbiting around the vortex line, which gives rise to the spin degrees of freedom. The geometric phase acquired by the scalar electron orbiting the vortex line induces the spin-orbit interaction. This leads to the fractional OAM states when we have a nonquantized monopole charge associated with the corresponding geometric phase. This involves a tilted vortex in EVBs. The monopole charge undergoes renormalization-group flow, which incorporates a length scale dependence making the fractional OAM states unstable upon propagation. It is pointed out that when EVBs move in an external magnetic field, the Gouy phase associated with the Laguerre-Gaussian modes modifies the geometric phase factor and a proper choice of the radial index helps to have a stable fractional OAM state.