  We have performed a microscopic study of a straight quantized vortex line in
three dimensions in condensed $^4$He at zero temperature using the Shadow Path
Integral Ground State method and the fixed-phase approximation. We have
characterized the energy and the local density profile around the vortex axis
in superfluid $^4$He at several densities, ranging from below the equilibrium
density up to the overpressurized regime. For the Onsager-Feynman (OF) phase
our results are exact and represent a benchmark for other theories. The
inclusion of backflow correlations in the phase improves the description of the
vortex with respect to the OF phase by a large reduction of the core energy of
the topological excitation. At all densities the phase with backflow induces a
partial filling of the vortex core and this filling slightly increases with
density. The core size slightly decreases for increasing density and the
density profile has well defined density dependent oscillations whose wave
vector is closer to the wave vector of the main peak in the static density
response function rather than to the roton wave vector. Our results can be
applied to vortex rings of large radius $R$ and we find good agreement with the
experimental value of the energy as function of $R$ without any free parameter.
We have studied also $^4$He above the melting density in the solid phase using
the same functional form for the phase as in the liquid. We found that
off-diagonal properties of the solid are not qualitatively affected by the
velocity field induced by the vortex phase, both with and without backflow
correlations. Therefore we find evidence that a perfect $^4$He crystal is not a
marginally stable quantum solid in which rotation would be able to induce
off-diagonal long-range coherence.
