Ginzburg-Landau theory of vortices in superfluid3He-B

Vortices in superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ differ qualitatively from all other vortices. In conventional quantized vortices ${(}^4$He, superconductors) the order parameter is a scalar function with cylindrical symmetry that vanishes on its axis, but in superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ the order parameter has nine complex components. More interestingly, the symmetries of the differential equation describing the vortices can be broken by its solutions. In the pressure-temperature phase diagram of ${}_{}{}^3{}_{}{}^{}\mathit{B}$, there is a first-order transition (observed in the shift of the NMR line) between two different vortices that, as recent theory has shown, have different broken symmetries. The vortex observed at high pressures has broken parity with either left- or right-handed vortices. The low-pressure vortex (which is the vortex obtained in the weak-coupling Bardeen-Cooper-Schrieffer theory) has, in addition, broken rotational symmetry around its axis, resulting in a novel double-core structure. The Ginzburg-Landau theory of vortices reproduces the properties of the transition, accounts well for the measured susceptibility and magnetization, and gives detailed predictions about properties not yet measured, such as the jump of the magnetization in the transition, the shift of the transition line in magnetic field, and the orientation of the double-core vortex. The lack of helical instability in the double-core vortex implies that the identification of the vortices is unique. The transition between vortices is interpreted in simple physical terms. A qualitative explanation is given to the metastable state of the low-pressure vortex observed at low temperatures. The numerical method for solving the Ginzburg-Landau equation is discussed.