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We study here fractional orbital angular momentum (OAM) states in electronvortex beams (EVB) from the perspective of geometric phase. We have consideredthe skyrmionic model of an electron, where it is depicted as a scalar electronorbiting around the vortex line, which gives rise to the spin degrees offreedom. The geometric phase acquired by the scalar electron orbiting aroundthe vortex line induces the spin-orbit interaction, which leads to thefractional OAM states with non-quantized monopole charge associated with thecorresponding geometric phase. This involves tilted vortex in EVBs. Themonopole charge undergoes the renormalization group (RG) flow, whichincorporates a length scale dependence making the fractional OAM statesunstable upon propagation. It is pointed out that when EVBs move in an externalmagnetic field, the Gouy phase associated with the Laguerre-Gaussian modesmodifies the geometric phase factor and a proper choice of the radial indexhelps to have a stable fractional OAM state.

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