  The spin-1/2 antiferromagnetic zig-zag ladder is studied by exact
diagonalization of small systems in the regime of weak inter-chain coupling. A
gapless phase with quasi long-range spiral correlations has been predicted to
occur in this regime if easy-plane (XY) anisotropy is present. We find in
general that the finite zig-zag ladder shows three phases: a gapless collinear
phase, a dimer phase and a spiral phase. We study the level crossings of the
spectrum,the dimer correlation function, the structure factor and the spin
stiffness within these phases, as well as at the transition points. As the
inter-chain coupling decreases we observe a transition in the anisotropic XY
case from a phase with a gap to a gapless phase that is best described by two
decoupled antiferromagnetic chains. The isotropic and the anisotropic XY cases
are found to be qualitatively the same, however, in the regime of weak
inter-chain coupling for the small systems studied here. We attribute this to a
finite-size effect in the isotropic zig-zag case that results from
exponentially diverging antiferromagnetic correlations in the weak-coupling
limit.
