  We use a combination of density matrix renormalization group calculations and
analytical approaches to study a simplified model for a spatially anisotropic
spin-1/2 triangular lattice Heisenberg antiferromagnet: the three-leg
triangular spin tube (TST). The model is described by three Heisenberg chains,
with exchange constant J, coupled antiferromagnetically with exchange constant
J' along the diagonals of the ladder system, with periodic boundary conditions
in the shorter direction. We determine the full phase diagram of this model as
a function of spatial anisotropy, J'/J, and magnetic field. We find a rich
phase diagram, which is dominated by quantum states - phases corresponding to
the classical ground state appears only in a small region. Among the dominant
phases generated by quantum effects are commensurate and incommensurate
coplanar quasi-ordered states, which appear in the vicinity of the isotropic
region for most fields, and in the high field region for most anisotropies. The
coplanar states, while not classical ground states, can be understood
semiclassically. Even more strikingly, the largest region of phase space is
occupied by a spin density wave phase, which has incommensurate collinear
correlations along the field. This phase has no semiclassical analog, and may
be ascribed to enhanced one-dimensional (1d) fluctuations due to frustration.
Cutting across the phase diagram is a magnetization plateau, with a gap to all
excitations and up up down spin order, with a quantized magnetization equal to
1/3 of the saturation value. In the TST, this plateau extends almost but not
quite to the decoupled chains limit. Most of the above features are expected to
carry over to the two dimensional system, which we also discuss. At low field,
a dimerized phase appears, which is particular to the 1d nature of the TST, and
which can be understood from quantum Berry phase arguments.
