Gapless excitations of axially symmetric vortices in systems with tensorial order parameter

We extend the results of previous work on vortices in systems with tensorial order parameters. Specifically, we focus our attention on systems with a Ginzburg–Landau free energy with a global U(1)_P×SO(3)_S×SO(3)_L$U{\left(1\right)}_P\times SO{\left(3\right)}_S\times SO{\left(3\right)}_L$ symmetry in the phase, spin and orbital degrees of freedom. We consider axially symmetric vortices appearing on the spin–orbit locked SO(3)_S+L$SO{\left(3\right)}_{S+L}$ vacuum. We determine the conditions required on the Ginzburg–Landau parameters to allow for an axially symmetric vortex with off diagonal elements in the order parameter to appear. The collective coordinates of the axial symmetric vortices are determined. These collective coordinates are then quantized using the time dependent Ginzburg–Landau free energy to determine the number of gapless modes propagating along the vortex.