Eilenberger equations for rotating superfluid ³he and calculation of the upper critical angular velocity Ω _c2

On the basis of Gorkov's formulation of superconductivity theory, generalized Eilenberger equations are derived which apply to rotating superfluid ³He in the presence of a magnetic field h and finite superflow v. In analogy to conventional type II superconductors, the possibility of vortex solutions is discussed. An implicit equation determining the upper critical angular velocity Ω_c2 as a function of temperature T, magnetic field h, and superflow Ν parallel to the rotation axis is·inferred from the linearized Eilenberger equations. In contrast to the case of slowly rotating ³He-A, the solution of the eigenvalue problem determining the order parameter δ near the upper critical angular velocity admits no coreless vortex solutions. The space-dependent amplitude of the order parameter is analogous to Abrikosov's vortex array solution, while the spin-orbit part is given either by a polar-state type or an Anderson-Brinkman-Morel (ABM)-state-type eigensolution. Among the possible eigensolutions the polar-state type yields for vanishing superflow v the highest critical rotation frequency. For finite superflow v parallel to the rotation axis, however, the ABM-state-type solution is stabilized in comparison to the polar state for |ν|≥0.2π(T_c0/T_F)_νF at zero temperature.