  Analytic properties of the multiple spin density waves showing vortex
structures, which have recently been found in the fcc Hubbard model using the
generalized Hartree-Fock (GHF) approximation, have been investigated. It is
shown that the 2Q multiple helical spin density waves (SDW) consist of two
types of half-skyrmion vortices and two types of antivortices. In the 3Q
multiple helical SDW, the half-skyrmion vortex structures with the "westerlies"
are shown to be twisted along the axis perpendicular to the vortex plane. The
12Q multiple SDW (12QMSDW) are analyzed as a superposition of the 2-4QMSDW on
the (001) plane and the remaining 4QMSDW. It is shown analytically that the
2-4QMSDW forms the planar vortices and antivortices with zero core
polarization, so that the 12QMSDW forms the same types of vortices with
additional SDW polarization in the [001] direction. These results are in good
agreement with the visualization image data based on the GHF calculations.
