Order parameters and energies of analytic and singular vortex lines in rotating³He-A

Abstract

We present the expressions of the generalized Ginzburg-Landau (GL) theory for the free energy and the supercurrent in terms of thed vector, the magnetic fieldH, and operators containing the spatial gradient and the rotationΩ. These expressions are then specialized to the Anderson-Brinkman-Morel (ABM) state. We consider eight single-vortex lines of cylindrical symmetry and radiusR=[2mΩ/ℏ]^−1/2: the Mermin-Ho vortex, a second analytic vortex, and six singular vortices, i.e., the orbital and radial disgyrations, the orbital and radial phase vortices, and two axial phase vortices. These eight vortex states are determined by solving the Euler-Lagrange equations whose solutions minimize the GL free energy functional. For increasing field, the core radius of the$\hat l$ texture of the Mermin-Ho vortex tends to a limiting value, while the core radius of the$\hat d$ texture goes to zero. The gap of the singular vortices behaves liker ^α forr → 0, where α ranges between$\sqrt {1/2}$ and$\sqrt {9/2}$. The energy of the radial disgyration becomes lower than that of the Mermin-Ho vortex for fieldsH≥6.5H*=6.5×25 G (atT=0.99T _c and forR=10L*=60 µm, orω=2.9 rad/sec). ForR → 2ξ _T (ξ _T is the GL coherence length) orω →ω _c2 (upper critical rotation speed), the energies of the singular vortices become lower than the energies of the analytic vortices. This is in agreement with the exact result of Schopohl for a vortex lattice atΩ _c 2. Finally, we calculate the correction of order (1 -T/T _c ) to the GL gap for the axial phase vortex.