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

Abstract

We extend the results of previous work on the vortex order parameter in systems similar to the Ginzburg-Landau description of superfluid $^3$He in the bulk B phase. Specifically, we consider vortices preserving an axial $U(1)$ symmetry. We determine the conditions required by the $\beta_i$ parameters to allow for an energetically favorable development of the off-diagonal antisymmetric and symmetric-traceless elements satisfying the axial symmetry from the trace-only ansatz of the order parameter. The number and type of gapless moduli appearing on the classical low energy theory of axial vortices is determined. The time-dependent part of the Ginzburg-Landau free energy is then considered to determine the number of quantized modes emerging from the gapless moduli fields.