Parts, Ü.; Karimäki, J.; Koivuniemi, J.; Krusius, M.; Ruutu, V.; Thuneberg, E.; Volovik, Grigory Phase diagram of vortices in superfluid 3He-A

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We discuss the general properties of periodic vortex arrangements in rotating superfluids. The different possible structures are classified according to the symmetry space-groups and the circulation number. We calculate numerically several types of vortex structures in superfluid 3 He-A. The calculations are done in the Ginzburg-Landau region, but the method is applicable at all temperatures. A phase diagram of vortices is constructed in the plane formed by the magnetic field and the rotation velocity. The characteristics of the six equilibrium vortex solutions are discussed. One of these, the locked vortex 3, has not been considered in the literature before. The vortex sheet forms the equilibrium state of rotating 3 He-A at rotation velocities exceeding 2.6 rad/s. The results are in qualitative agreement with experiments. Some of the theoretical results that we presented in Refs. 1,2 were found to be incorrect in further calculations. These errors are corrected here. As a consequence, the present phase diagram of vortices differs form the one in Refs. 1,2. In particular, there appears a new vortex structure, the locked vortex 3, but also the locations of other phase boundaries are changed.For introduction to superfluid 3 He 3,4 and its vortices 5-9,2 we refer to various review articles. Although not introductory, this paper intends to be a complete exposition of what is needed for understanding the equilibrium vortex structures in bulk superfluid 3 He-A.We start in Section I with the formulation of the vortex problem, which is general enough for all superfluids and can be generalized also to superconductors. This gives a general classification of vortex states based on spacegroup symmetry and circulation number. The classification is continued in Section II using properties specific to 3 He-A. The calculations of the vortex structures are based on the hydrostatic theory, which is discussed in Section III, and the calculational method is described in Section IV. Detailed description of the different vortex types is given in Section V. The correspondence with experiments is discussed in Section VI. I. THE GENERAL VORTEX PROBLEMLet us consider an uncharged fluid (in practice 4 He or 3 He) in a container rotating at angular velocity Ω. We will neglect all complications arising from the finite size of the container. Although we will not discuss the detailed correspondence, the analysis in this section is also applicable to a charged fluid (superconductor) when Ω is replaced by the averaged magnetic field B.At the microscopic level, the fluid has the effective Hamiltonian H eff = H 0 − Ω · J. Here H 0 = i (p 2 i /2m) + V is the Hamiltonian in a nonrotating system, which consists of a kinetic energy term and an interaction energy term V . The angular momentumconsists of an orbital and a spin part. We can write H eff in the formwhere v n,i = Ω × r i is the "normal fluid" velocity at the location of the particle i. The last term is the centrifugal energy. It causes the pressure to increase with increasing distance from ...

We study numerically the behavior of a single quantized vortex in a rotating cylinder. We study in particular the spiraling motion of a vortex in a cylinder that is parallel to the rotation axis. We determine the asymptotic form of the vortex and its axial and azimuthal propagation velocities under a wide range of parameters. We also study the stability of the vortex line and the effect of tilting the cylinder from the rotation axis.Comment: 9 pages, 10 figures. Considerable changes, now close to the published versio